Optoelectronics of Molecules and Polymers (Springer Series in Optical Sciences)

Springer Series in OPTICAL SCIENCES Founded by H. K.V. Lotsch Editor-in-Chief: W.T. Rhodes, Atlanta Editorial Board : ...

Author: Andre Moliton

65 downloads 1194 Views 25MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

Springer Series in

OPTICAL SCIENCES Founded by H. K.V. Lotsch Editor-in-Chief: W.T. Rhodes, Atlanta Editorial Board : T. Asakura, Sapporo T.W. Hansch , Garching T. Kamiya, Tokyo F. Krausz, Garching B. Monemar, Linkoping H. Venghaus, Berlin H. Weber, Berlin H. Weinfurter , Mtinchen

104

Springer Series in

OPTICAL SCIENCES The Springer Series in Optical Sciences, under the leadership of Editor-in-Chief William T. Rhodes, Georgia Institute of Technology, USA, provides an expanding selection of research monographs in all majorareas of optics: lasers and quantum optics, ultrafast phenomena, optical spectroscopy techniques, optoelectronics, quantum information, information optics, applied laser technology, industrial applications, and other topics of contemporary interest. With this broadcoverage of topics, the series is of use to all research scientists and engineers who need up-to-date reference books. The editors encourage prospective authors to correspond with them in advance of submittinga manuscript. Submission of manuscripts should be madeto the Editor-in-Chief or one of the Editors. See also www.springeronline.comlseries/624 Editor-in-Chief

William T. Rhodes Georgia Institute of Technology School of Electricaland Computer Engineering Atlanta, GA 30332-0250, USA E-mail: [email protected]

Editorial Board

Toshimitsu Asakura

Bo Monemar

Hokkai-Gakuen University Faculty of E~gineering I-I, Minami-26, Nishi II, Chuo-ku Sapporo, Hokkaido064-0926,Japan E-mail: [email protected]

Department of Physics and Measurement Technology MaterialsScience Division Linkoping University 58183 Linkoping, Sweden E-mail: [email protected]

Theodor W. Hansch Max-Planck-Institut fur Quantenoptik Hans-Kopfennann-StraBe I 85748 Garching, Gennany E-mail: [email protected]

Herbert Venghaus Heinrich-Hertz-Institut fOr Nachrichtentechnik Berlin GmbH Einsteinufer 37 10587Berlin. Gennany E-mail: venghaus@hhLde

Takeshi Kamiya Ministryof Education, Culrure, Sports Science and Technology National Institutionfor Academic Degrees 3-29-1 Otsuka, Bunkyo-ku Tokyo 112-0012, Japan E-mail: [email protected]

Ferenc Krausz Ludwig-Maximilians-Universitllt MUnchen Lehrstuhl fOr Experimentelle Physik Am Coulombwall I 85748 Garching, Gennany and Max-Planck-Institut fUr Quantenoptik Hans-Kopfennann-StraBe 1 85748 Garching, Gennany E-mail:[email protected]

Horst Weber TechnischeUniversitat Berlin OptischesInstitut StraBe des 17. Juni 135 10623Berlin, Gennany E-mail: [email protected]

Harald Weinfurter Ludwig-Maximilians-Universitlit Munchen Sektion Physik SchellingstraBe 4/1II 80799 MUnchen, Gennany E-mail: [email protected]

Andre Moliton

Optoelectronics of Molecules and Polymers With 229 Illustrations

~ Springer

Roger C: Hiorns (Translator) Laboratoire de Physico-Chimie des Polymeres, Universite de Pau et des Pays de I' Adour, France [email protected]

Andre Moliton Unite de Microelectronique, Optoelectronique et Polyrneres, Universite de Limoges France amoliton @unilim.fr

Library of Congress Control Number: 2005925187 ISBN-IO: 0-387-23710-0 ISBN-13: 978-0387 -23710-7

e-ISBN: 0-387-25103-0

Printed on acid-free paper. © 2006 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 100I3, USA), except for brief excerpts in connection with reviews or scholarly analysis . Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America.

98765432 I springeronline.com

(TechsellEB)

To Colette, Celine and Vivien

To the memory of my mother and father

Preface

Molecular materials have been known since the 1960's to show semiconducting properties, but the past decade has seen an unprecedented level of activity. This has been driven both by scientific advance , and also by the prospects for new technologies based on large-area deposition of thin-film organic semiconductors. Molecular semiconductors have come of age, and those active in the field now need access to a modern account of the semiconductor physics and engineering that underpins the emerging applications of these materials including light-emitting diodes for displays , photovoltaic diodes, and electro-optical modulators. The necessity of bringing together realisable technology, which is primarily the process of thin-film deposition of semiconductor films, with useful semiconductor properties still intact often produces a complex physical system. In particular, disorder is inevitably present in these structures, and one of the central issues, which must determine the limitations for useful applications, is the way in which disorder modifies electronic structure. At its most simple, disorder in molecular materials is much less damaging to semiconducting properties than is the case in inorganic semiconductors. Deviation from perfect crystalline order in inorganic semiconductors generally produces broken chemical bonds which produce 'non-bonding' energy levels within the semiconductor gap, and this is particularly important at interfaces. However, disorder in molecular materials can often be accommodated without the breaking of chemical bonds, so that surfaces or interfaces can also, in this sense, remain chemically intact. In this situation, working semiconductor devices, such as LEDs, can operate when made from very disordered layers of molecular semiconductor. However, more careful examination of the role of disorder is required, because it affects electronic charge transport very strongly, and the understanding of how this controls device operation is of great importance. Andre Moliton has written the book that provides students and researchers in this field with the broad perspective that is now needed . He provides a very clear picture of the way in which molecular semiconductors derive their semiconducting properties, and he develops the description of their electronic transport and optical properties which derive from the molecular characteristics of these materials, such as weak delocalisation of electron states, and local vibration coupling, leading to polaron

viii

Preface

formation . His account is particularly useful in bringing his knowledge from the field of inorganic disordered semiconductors, such as amorphous silicon, to provide a useful model for understanding of the role of disorder here . The integration of this with the operation of semiconductor device structures provides the main theme for the later part of this book, and many students will find this particularly useful. In summary, I commend this book for bringing together with great clarity : the framework of traditional semiconductor science , the new physics which comes with molecular semiconductors, and the complexities associated with real semiconductor device fabrication , particularly disorder. It will be much appreciated by both experts and students alike. Sir Richard Friend Cavendish Professor of Physics Cambridge November 2002

Abbreviations

Alq3

CB

CP CuPc £r

EL ET FB FET

HOMO I

ITO Js Jsp

LED LPPP LUMO M

MIM MIS MT OLED PEDOT PLED PMMA

tris(8-hydroxyquinoline) aluminium conduction band conducting polymer copper phtalocyanine relative dielectric permittivity electroluminescence electronic tran sver se mode forbidden band field effect transistor highest occupied molecular orbital insulator indium tin oxide saturation current saturation current in the presence of trap s light emitting diode ladder poly(para-phenylene) lowe st unoccupied molecular orbital metal metal-insulator-metal metal-insulator-semiconductor magnetic transverse mode organic light emitting diode poly(3 ,4-eth ylenedioxythiophene) polymer light emitting diode poly(methyl methacrylate)

PPP

poly(para -phenylene)

PPV PVK

poly(para-phenylene vinylene) polyvinylcarbazole semiconductor space charge limited current thermoelectric power trap -filled-limit valence band tension at which all trap s are filled

SC SCLC TEP TFL

VB VTFL

Introduction: the origin and applications of optoelectronic properties of molecules and polymers

During the second half of the twentieth century, an upheaval in industrialised societies has been brought about by the application of the properties of certain solids to electronic and optical fields. Solids recognised as semiconductors have caused the greater part of this change. Even while the band theory for silicon was incomplete [fis 88], radio valves were being replaced by transistors at the end of the 1950s. The electronic industry then benefited from developments in solid state theories , which yielded a more thorough understanding of properties specific to semiconductors. Simultaneously, innovative techniques, such as wafer technology, permitted miniaturisation of electronic components. Their 'integration' into circuits gave the computer revolution, which we are still witnessing. Central to this technology is the mobility that electrons display in perfect crystals-around 104 crrr' V-I s-I. There remains a large number of devices for which this rate of commutation and response time is no where as near important. This can be found in a wide range of display technologies, for example in standard video technology where an image is replaced once every 1/ 25th of a second , and also in photovoltaic systems, in which the goal is to separate and collect photogenerated charges without paying any particular attention to the duration of processes brought into play. In such systems, the desire for the properties which crystalline semiconductors do not have is very much in evidence. It includes the need for mechanical flexibility and low cost fabrication of devices covering large surface areas . After having equipped the American satellite Vanguard with solar cells made from crystalline silicon to provide power for its radio, research was initiated into amorphous silicon to fulfil these design requirements. The initial conquest of space was then followed by the use of satellites which, with their total of hundreds of thousands of solar cells, now allow communication by telephone and television between continents. In reality, the development of amorphous silicon was troubled by the long thought theory that amorphous semiconductors could not be doped . The theory at the time was based on the '8-N' rule (N is the number of electrons in the outer layer of a given atom), which was applied to glass-like materials and supposed that each atom was surrounded by 8-N close neighbours. All electrons could then be thought of being 'engaged' either in bonds between neighbouring atoms, or in 'isolated pair' orbitals.

xii

Introduction

This would disallow the possibility of having free carriers coming from incomplete bonds. However, amorphous silicon is not a glass in proper terms and it does not always follow the 8-N rule. For example, phosphorus atoms which often exhibit a co-ordination number of 3, and are thus not electronically active as ascribed by the 8-N rule, can take on a co-ordination number of 4, just as in the crystalline form . The 5th phosphorus electron does not therefore remain on the phosphorus atom , but falls into a dangling silicon bond site-a bond characteristic of the amorphous state and associated with a state localised within the so-called mobility gap. The Fermi level, initially situated more or less at the centre of the gap is displaced towards the bottom of the conduction band (Ec) by a quantity ~EF , which is of the order of ND/N (EF) where ND denotes the electron density coming from the donor phosphorus atoms and N (EF) represents the state density function for states close to the Fermi level. With a low value of N(EF), which can be realised by saturating the dangling silicon bonds with hydrogen, the displacement of the Fermi level can become significant and, in consequence, the doping effect becomes all important. Even though the properties of amorphous semiconductors such as hydrogenated silicon (a-Si:H) are well understood [com 85], their use remains relatively limited. This is because of two factors. The first is that the properties of these materials are inferior to those of crystalline silicon-for examples, they exhibit mobilities of the order of 1 cm 2 V-I s-I and, in photovoltaic cells, have efficiencies of around 14 % against 24 % for the crystalline material. Secondly, their manufacture remains difficult and expensive, as they require techniques involving deposition under vacuum or chemical reactions in the vapour phase and the use of high temperatures. In 1999, photovoltaic cells fabricated using amorphous silicon accounted for only 12 % of total production. All things are not bright for crystalline silicon though. This material remains expensive, and the photovoltaic industry requires cheaper material from a pathway which does not involve the costly wastes associated with selecting the best parts of silicon ingots usually fabricated for the electronics industry. Against this background, the possibility that organic materials, which might be facile to manipulate and relatively cheap to obtain, has always held a certain interest. A serious brake on their use though has been the poor understanding of the origins of their electronic and optical properties. In addition, as much as for amorphous inorganic semiconductors, it has taken considerable time to master their n- and ptype doping, an absolute necessity for their use in devices . In contrast to inorganic semiconductors, organic semiconductors are not atomic solids : they are n-conjugated materials in which the transport mechanisms vary considerably from those classically derived for solids. And even more so than in inorganic materials, the roles of defaults such as traps and structural inhomogeneities play an essential role in determining the transport and interstitial and non-substitutional doping phenomena. Polymers which have conjugated double bonds yielding rt-conjugation present two particular advantages. They exhibit good electronic transport properties-with a facile delocalisation of electrons-and good optical properties. The energy separation between n- and rr"-bands (which resemble, respectively, valence and conduction bands for solid state physicists), is typically around 1.5 to 3 eV, a value ideal for the optical domain. It is worth noting that polymers such as polyethylene, which contain

Introduction

xiii

only single o-bonds, exhibit high resisitivities due to the localisation of electrons around o-bonds. Due to transitions between o- and o"-bands of the order of 5 eY, the properties of these materials lie beyond the optical domain. Certain rr-conjugated materials have been studied throughout the 20th century. Anthracene is a particularly good example [pop 82], as quantum theories developed for this material allowed an interpretation of its numerous physical properties. The large size of the crystals obtained necessitated extremely elevated tensions to study their electroluminescence, arresting initial developments. Similarly,linear polyacetylenes, firstprepared in 1968[ber 68], were only available as black powders,a near impractical form for characterising. The fabrication of organic materials in thin films permitted a developmentof the number of derivedapplications. Over the last few decades, decisive developments have occurred: the preparation of polyacetylene in film form, realised in 1977 by A. MacDiarmid and H. Shirakawa which gained them and the physicist A. Heeger the Noble prize for chemistry in 2000; and the improved understanding of transport and doping properties of rr-conjugated materials [su 79 and bre 82b]. The insertion of these materials into devices such as electroluminescent diodes made sure of their important position in the optoelectronics industry of the 21st century. There are notable articles which come from this period and include for 'small molecules' that published in 1987on Alq3 [tan 87], and for n-conjugated polymers that published in 1990 on poly(phenylene vinylene) (PPY) [bur 01]. Nearly a century after the initial discovery of photoconductivity in anthracene in 1906 by Pocchetino, we have realised the 'dream' of optoelectronic and electronic technology based on thousands and even millions [pop 82] of organic molecules and polymers. All this has been made possible by synthetic organic chemistry. As will be detailed in this book, it is now possible to adjust the optical gap and thus the absorption and emission wavelengths of materials simply, for example, by adjusting organic, molecular chain lengths. These organic materials have exhibited not only photoconductivity and electroluminescence but also triboelectric properties, metallike conductivity, superconductivity, photovoltaic effects, optical and non-linear effects and even charge stocking and emission. Elaborated theories have demonstrated that the electronic and optical properties of organic solids are intimately tied, just as for inorganic materials. These theories are based on interpretations of the quantum mechanics and the macroscopic environment, including effects caused by impurities and defaults, of the molecules. Quasi-particles, such as polarons and excitons, introduced through the understanding of inorganic structures, are also useful in explaining electronic and optical behaviours of organic materials, although they present particularities specific to the organic media such as non-negligible electron-lattice interactions. This book aims to bring together the practically indispensable knowledge required to understand the properties specific to organic devices which chemists, physicists and specialists in electronics are now proposing. Accordingly, after discussing concepts used for well organised solids, notably for one-dimensional systems, we detail the concepts used for localised levels in disordered materials (initially developed for amorphous inorganic solids). The first approximations to real organic materials are then made [mol 91 and mol 96], which have recently been further developed by

xiv

Introduction

Reghu Menon, Yoon, Moses and Heeger [men 98]. After having passed by the requisite step of determining energy levels in 'perfect' molecular and macromolecular systems , the effect of quasi-particles are introduced. The importance of the latter type of particles is stressed, as their role in the interpretation of transport or doping mechanisms and in optical processes is singularly important. The role of excitons-here an electronic excitation in an organic solid-is detailed, and thus the transfer mechanism of an excitation between two interfaces is studied. The role of aggregates in solid matrices is also explored with respect to understanding charge injection , resistive media and the problems encountered at interface s. Technological developments are discussed in the second part of this book . The practicalities of fabricating devices using dry etching techniques with ion beams for example are presented. This information is supplemented by descriptions of the optoelectronic characterisations which can be performed. With the exception of the brief Appendix A-II, which is more a catalogue of the most important organic materials , the organic synthesis of these materials is not touched upon, and the reader in search of this information can look to some excellent reviews such as [bar OOa, den 00 and kra 98]. Another reason is of course that organic synthesis does not make up part of the competencies of the author! The devices based on organic materials that are covered are those which, for the present time, monopolise the attention of many researchers in the fields of their application. Included are the domains of visualisation, photon -electron conversion (photovoltaics) and electro-optical modulation which essentially concerns communications . These domains are often interdependent: we have already mentioned that communication satellites often require the use of photovoltaic cells. Indeed the insertion ofsuch cells into portable telephones and computers would often resolve problems associated with the lifetimes of batterie s. We can imagine that the propertie s of these materia ls has been treated at different scales :

+ Molecular scale At the scale of the individual molecule

+

Molecular electronic s (for example , using nano-tubes)

+

+

Mesoscopic scale

Macroscop ic scale

micelle s, vesicles membr anes Molecu lar films: mono layers (and Langmuir-Blodgett films)

Solid molecul ar mater ials

" t "-

Electronic

Photonic

(magnet s, ferroelectrics, superconductors)

The macroscopic scale is the oldest area of study and the closest to the realm of applications. And it is in this area that this book attemp ts to place active, n-conjugated materials which can serve as: • • • •

conductors or semiconductors of electricity; electro luminescent emitters ; 'converters' of light into electricity ; and optically active devices, and in particular, second-order non-linear devices.

Introduction

xv

More precisely, we will encounter the use of organic material s in the following components: •

•

•

light emitting devices, or rather, organic light emitting devices (OLEDs) or polymer light emitting devices (PLED s). The whole of Chapter 10 is dedicated to the various uses of these LEDs in displays and include s details on one of the first organic screens developed for commercial use in portable telephones. Many industrial groups own prototypes of this sort of component (see the conclusion of Chapter 10). Lasing diodes , diodes which emit white light or infra-red are also presented along with their applications; photovoltaic cells , where we will detail the originality of the organic based structures currently under development and based on interpenetrating networks of donor and acceptor molecules. While organic materials posses the advantage of increasing electron mobilities with temperature, they present-for the meantime-problems of stability ; and electro-optical modulators (of phase or amplitude) where the problems of handling and fabrication into devices and the choice of appropriate materials is discussed. The high resistance of certain polymers, such as polyimides, to oxygen and humidity does leave open the possibility of fabricating low cost device s which could be installed without precaution, perhaps underground, in both commercial and domestic environments.

This book was written with the desire to make it 'self-supporting' so that the reader would not find it necessary to turn to numerous, specialised publications and manuals . Appendices which give further details have been added and consist of 'aide-memoires' and supplementary information, and can give answers to albeit less fundamental but just as important questions. These Appendices may be of particular use to readers more used to books on physic s, electronics, physical-chemistry and chemistry. This manual is destined to be used by students following their first degree , but was especially written for Masters and PhD students. It should give a solid introduction to the optoelectronics of organic solids to researchers and engineers who wish to be more involved in the field, and I have no doubt that they will be passionate follows of a technology undergo ing full expansion.

Contents

Preface by Richard H. Friend. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Xl

Part One: Chapter I:

Concepts: Electronic and optical processes in organic solids Band and electronic structures in regular J-dimensional media .

II

III

IV

An introduction to approximations of weak and strong bonds. .... ...... . . .. ..... . . . . . . . . . . . . . . . . . . . .. . . I Materials with weak bonds . . . . . . . . . . . . . . . . . . . . 2 Materials with strong bonds . . . . . . . . . . . . . . . . . . . Band Structure in weak bonds .. . . . . . . . . . . . . . . . . . . . . I Prior result for zero order approximation 2 Physical origin of forbidden bands . . . . . . . . . . . . . . 3 Simple estimation of the size of the forbidden band. . .. . . .. . . . . . . . . . . . Floquet's theorem : wavefunctions for strong bond s. . . . . I Form of the resulting potential . . . . . . . . . . . . . . . . . 2 The form of the wavefunct ion . . . . . . . . . . . . . . . . . . 3 Floquet's theorem : effect of potential periodicity on wavefunction form . . . . . . . . . . . . . . . . . . . . . . . . A study on energy I Defining equations (with X == r: I - D) . . . . . . . . . . 2 Calculation of energy for a chain of N atoms . . . . . 3 Addit ional comments: physical significance of terms (Eo - a) and ~; simple calculation of E; and the appearance of allowed and forbidden bands in strong bonds. . . . . . . . . . . . . . . . . . . . . . . . .

3 3 3 4 6 6 6 8 9 9 IO II 12 12 13

16

xviii Contents V

I-D crystal and the distorted chain. .. .. . . .. .. . . ... . . 1 AB type crystal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The distorted chain VI Density function and its application, the metal insulator transition and calculation of Erel ax . . . . . . . . . . . . . . . . . . . I State density functions 2 Filling up zones and Peierls insulator-metal transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Principle of the calculation of Erel ax for a distorted chain . . .. .. . . . .. . . . .. .. . . . .. .. . . . . . . .. . . .. . VII Practical example: calculation of wavefunction energy levels, orbital density function and band filling for a regular chain of atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . I Limits of variation in k . . . . . . . . . . . . . . . . . . . . . . . 2 Representation of energy and the orbital density function using N = 8 . . . . . . . . . . . . . . . . . . . . . . . . . 3 Wavefunction forms for bonding and antibonding states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Generalisation regarding atomic chain states .. . . . VIII Conclusion. ... .. . .. .. . ..... . . .... . . . ... . ... . .. ..

Chapter II:

Electron and band structure . . . . . . . . . . . . . . . . . . . . . . . . . .. I Introduction .. . . .. . . . II Going from l-D to 3-D . . " . . . . . . . 1 3-D General expression of permitted energy. . . . .. 2 Expressions for effective mass, band size and mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III 3-D covalent crystal from a molecular model: sp3 hybrid states at nodal atoms . . . . . . . . . . . . . . . . . . . . . . . . 1 General notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Independent bonds: formation of molecular orbitals 3 Coupling of molecular orbitals and band formation IV Band theory Iimts and the origin of levels and bands from localised states 1 Influence of defaults on evolution of band structure and the introduction of 'localised levels' . 2 The effects of electronic repulsions, Hubbard's bands and the insulator-metal transition 3 Effect of geometrical disorder and Anderson localisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. V Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

19 19 20

22 22 24 25

26 26 26 27 30 30 33 33 34 34 35 36 36 38 40 41 41 43 47 57

Chapter III:

Contents

xix

Electron and band structures of 'perfect' organic solids. ..

59 59 59 59

I

II

III

IV

Chapter IV:

Introduction: organic solids 1 Context . ... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Generalities. . . . .. . . . . .. .. .. .. .. .. . .. . .. . ... . 3 Definition of conjugated materials; an aide-memoire for physicians and electricians . . . .. Electronic structure of organic intrinsic solids : n-conjugated polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Degenerate n-conjugated polymers. . . . . . . . . . . . . 2 Band scheme for a non-degenerate n-conjugated polymer: poly(para-phenylenc) . . . . . . . . . . . . . . .. Electronic structure of organic intrinsic solids: small molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of energy levels in going from an isolated chain to a system of solid state condensed molecules 2 Energy level distribution in Alq3 . . . . . . . . . . . . . .. 3 Fullerene electronic levels and states. . . . . . . . . . .. Conclusion : energy levels and electron transport . . . . . . .

63 63 65 68

68 69 70 74

Electron and band structures of 'real' organic solids. . . . .. I II

77 Introduction: 'real' organic solids 77 Lattice-charge coupling-polarons . . . . . . . . . . . . . . . . . . 77 I Introduction . ... ... . . . .. . . . . . .. . . . . . . . . . . . . . 77 2 3

III

IV

Chapter V:

62

Polarons . . .. . . . . . . . . . . . . . .. . . . . . .. . . . . ... . . Model of molecular crystals. . . . . . . . . . . . . . . . . .. 4 Energy spectrum of small polaron . . . . . . . . . . . . . . 5 Polarons in rr-conjugated polymers. . . . . . . . . . . . . 6 How do we cross from polaron-exciton to polaron ? 7 Degenerate n-conjugated polymers and solitons .. Towards a complete band scheme I Which effects can intervene? 2 Complete band scheme accumulating different pos sible effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Alq3 and molecular crystals . . . . . . . . . . . . . . . . . . . Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78 79

83 85 87 88 90 90 91 93 95

Conduction in delocalised, localisedand polaronic states . . 99 I II

Introduction . . . . . . . . . . . . . . . . . . . . . . . . .. 99 General theories of conduction in delocalised states . . . . 100 General results of conductivity in a real crystal : limits of classical theories. . . . . . . . . . . . . . . . . . . . . 100

xx

Contents 2

III

IV

V

VI

VII

Chapter VI:

Electrical conduction in terms of mobilities and the Kubo-Greenwood relationship: reasoning in reciprocal space and energy space for delocalised states Conduction in delocalised band states: degenerate and non-degenerate organic solids I Degenerate systems . . . . . . . . . . . . . . . . . . . . . . . . .. 2 Non-degenerate systems: limits of applicability of the conduction theory in bands of delocalised states for systems with large or narrow bands (mobility condition) . . . . . . . . . . . . . . . . . . . . . . . . .. Conduction in localised state bands. . . . . . . . . . . . . . . . . . 1 System 1: Non-degenerated regime; conductivity in the tail band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 System 2: degenerate regime; conductivity in deep localised states Transport mechanisms with polarons I Displacements in small polaron bands and displacements by hopping . . . . . . . . . . . . . . . . . . . . . 2 Characteristics of hopping by small polarons . . . . . 3 Precisions for the 'semi-classical' theory: transition probabilities. . . . . . . . . . . . . . . . . . . . . . .. 4 Relationships for continuous conductivity through polaron transport 5 Conduction in 3D in rr-conjugated polymers Other envisaged transport mechanisms . . . . . . . . . . . . . . . I Sheng's granular metal model 2 Efros-Shklovskii's model from Coulombic effects 3 Conduction by hopping from site to site in a percolation pathway . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Kaiser's model for conduction in a heterogeneous structure Conclusion: real behaviour. . . . . . . . . . . . . . . . . . . . . . . . . 1 A practical guide to conducting polymers . . . . . . . . 2 Temperature dependence analysed using the parameter w = -[(a In p)/a In T]

101 103 103

105 109 110 111 116 116 II? 120 122 124 128 128 128 128 129 129 129 131

Electron transport properties . . . . . . . . . . . . . . . . . . . . . . . . . . 133 I II

III

Introduction Basic mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Injection levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Three basic mechanisms . . . . . . . . . . . . . . . . . . . . . . Process A: various (emission) currents produced by electrodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

133 133 133 134 135

Contents

IV

V

VI

I Rectifying contact (blocking metal ---+ insulator) . . 2 Thermoelectronic emission (T =1= 0; E, = 0) . . ... 3 Field effect emission (Shottky): E, is 'medium intense' .......................... 4 Tunnelling effect emissions and Fowler-Nordheim's equation Process B (simple injection): ohmic contact and current limited by space charge. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Ohmic contact (electron injection) 2 The space charge limited current law and saturation current (Js) for simple injection in insulator without traps 3 Transitions between regimes. . . . . . . . . . . . . . . . . .. 4 Insulators with traps and characteristics of trap levels 5 Expression for current density due to one carrier type (Jsp) with traps at one discreet level (Ed; effective mobility . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6 Deep level traps distributed according to Gaussian or exponential laws Double injection and volume controlled current: mechanism C in Figure VI-2 I Introduction: differences in properties of organic and inorganic solids 2 Fundamental equations for planar double injection (two carrier types) when both currents are limited by space charge : form of resulting current JyCC (no trap nor recombination centre s) 3 Applications .. .. . . . . . . . . . .. . .. .. . .. . .. . . . . . . The particular case of conduction by the Poole-Frenkel effect I Coulombic traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Conduction due to Poole-Frenkel effect (as opposed to Schottky effect) . . . . . . . . . . . . . . . . . . ..

Chapter VII: Optical processes in molecular and macromolecular solids I Introduction II Matrix effects due to insertion of atoms with incomplete internal electronic levels. . . . . . . . . . . . . . . . . . . . . . . . . . . Electronic configuration of transition elements and rare earths 2 Incorporation of transition metals and rare earths into dielectric or a semiconductor matrix : effects on energy levels

xxi 135 136 136 137 138 138

139 143 144

147 151 154 154

155 157 159 160 160 163 163 164 164

165

xxii

Contents 3

III

IV

V

VI

Transitions studied for atoms with incomplete layers inserted in a matrix Classic optical applications using transition and rare earth elements 1 Electroluminescence in passive matrices. . . . . . . . . 2 Insertion into semiconductor matrix 3 Light amplification: erbium lasers . . . . . . . . . . . . . . Molecular edifices and their general properties 1 Aide memoire : basic properties. . . . . . . . . . . . . . .. 2 Selection rule with respect to orbital parities for systems with centre of symmetry . . . . . . . . . . . . . . . 3 More complicated molecules: classical examples of existing chromophores . . . . . . . . . . . . . . . . . . . . . Detailed description of the absorption and emission processes in molecular solids. . . . . . . . . . . . . . . . . . . . . . . 1 Electron-lattice coupling effects during electron transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Selection rules and allowed transitions . . . . . . . . .. 3 Modified Jablonsky diagram and modification of selection rules: fluorescence and phosphorescence 4 Experimental results: discussion Excitons 1 Introduction.. . .. . . . .. . . . . . .. .. . . . . . . . . . . . . . 2 Wannier and charge transfer excitons 3 Frenkel excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 States, energy levels and transitions in physical dimers 5 System containing an infinite number of interacting molecule s and exciton band: Davidov displacement and breakdown 6 Aggregates... .. ... . . . . . . . ....... . . . . . . . . . . . 7 Forster and Dexter mechanisms for transfer of electron excitation energy . . . . . . . . . . . . . . . . . . . ..

167 171 171 172 173 174 174 176 177 179 179 180

181 183 185 185 186 188 189

192 194 195

Part Two: Components: OLEDs, photovoltaic cells and electro-optical modulators Chapter VIII: Fabrication and characterisation of molecular and macromolecular optoelectronic components 1

201 Deposition methods 201 1 Spin coating 201 2 Vapour phase deposition . . . . . . . . . . . . . . . . . . . . . . 202

Contents xxiii 3

II

III

IV

Chapter IX:

Polymerisation in the vapour phase (VDP method) 4 Film growth during vapour deposition: benefits due to deposition assisted by ion beams 5 Comment: substrate temperature effects . . . . . . . . . Fabrication methods : OLEDs and optical guides for modulator arms 1 OLED fabrication 2 Fabrication of modulator guides/arms from polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photometric characterisation of organic LEDs (OLEDs or PLEDs) 1 General definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Internal and external fluxes and quantum yields : emissions inside and outside of components . . . . .. 3 Measuring luminance and yields with a photodiode Characterisation of polymer based linear wave guides .. 1 Measuring transversally diffused light . . . . . . . . . .. 2 Loss analyses using 'Cut - Back' and ' Endface Coupling ' methods

Organic structures and materials in optoelectronic emitters I Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. II How CRTs work III Electroluminescent inorganic diodes. . . . . . . . . . . . . . . . . I How they work , 2 Display applications 3 Characteristic parameters 4 In practical terms. . . . . . . . . . . . . . . . . . . . . . . . . . . . IV Screens based on liquid crystals . . . . . . . . . . . . . . . . . . . . . I General points 2 How liquid crystal displays work . . . . . . . . . . . . . . . 3 LCD screen structure and the role of polymers . . . 4 Addressing in LCD displays . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . .. .. ... .. ... . . . . . . .. . . ..... V Plasma screens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. VI Micro-point screens (field emission displays (FED» . .. VII Electroluminescent screens . . . . . . . . . . . . . . . . . . . . . . . . . 1 General mechanism . . . . . . . . . . . . . . . . . . . . . . . . .. 2 Available transitions in an inorganic phosphor .. . . 3 Characteristics of inorganic phosphors from groups II-VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

203 204 209 210 210 212 217 217 221 226 232 232 233

235 235 235 236 236 237 237 238 239 239 240 242 243 244 244 245 246 246 247 249

xxiv Contents 4

Electroluminescent think film displays: how they work with alternating currents . . . . . . . . . . . . . . . .. 5 Electroluminescent devices operating under direct current conditions VIII Organic (OLED) and polymer (PLED) electroluminescent diodes . . . . . . . . . . . . . . . . . . . . . . . . .. I Brief history and resume. . . . . . . . . . . . . . . . . . . . .. 2 The two main developmental routes 3 How OLEDs function and their interest

Chapter X:

Electroluminescent organic diodes. . . . . . . . . . . . . . . . . . . . . . I Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II Comparing electronic injection and transport models with experimental results 1 General points: properties and methods applied to their study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2 Small molecules (Alq3) . . . . . . . . . . . . . . . . . . . . . . . 3 Polymers.... . . .. .. .... .. . .. .... .. .. .... . .. . III Strategies for improving organic LEDs and yields . . . . . . I Scheme of above detailed processes 2 Different types of yields . . . . . . . . . . . . . . . . . . . . .. 3 Various possible strategies to improve organic LED performances IV Adjusting electronic properties of organic solids for electroluminescent applications . . . . . . . . . . . . . . . . . . . .. I A brief justification of n- and p-type organic conductivity 2 The problem of equilibrating electron and hole injection currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Choosing materials for electrodes and problems encountered with interfaces 4 Confinement layers and their interest V Examples of organic multi-layer structures I Mono-layer structures and the origin of their poor performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The nature of supplementary layers. . . . . . . . . . . . . 3 Classic examples of the effects of specific organic layers 4 Treatment of the emitting zone in contact with the anode VI Modification of optical properties of organic solids for applications I Adjusting the emitted wavelength 2 Excitation energy transfer mechanisms in films doped with fluorescent or phosphorescent dyes . ..

250 251 253 253 253 254 257 257 258 258 259 267 272 272 273 274 276 276 277 277 279 279 279 280 280 284 285 285 286

Contents Circumnavigating selection rules: recuperation of non-radiative triplet excitons 4 Energy transfer with rare earths and infrared LEDs 5 Microcavities .. . . . . .... . . . . . . . . . . . . . . . . . .. . . 6 Electron pumping and the laser effect . . . . . . . . . .. VII Applications in the field of displays: flexible screens . .. I The advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2 The problem of ageing 3 The specific case of white diodes. . . . . . . . . . . . . . . 4 The structure of organic screens . . . . . . . . . . . . . . . . 5 A description of the fabrication processes used for organic RGB pixels . . . . . . . . . . . . . . . . . . . . . .. 6 Emerging organic-based technologies : flexible electronic 'pages' VIII The prospective and actual production at 2002 IX Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. X Actual state-of-the-art and prospectives

xxv

3

Chapter XI:

Organic photovoltaic devices I Principles and history of organic based photovoltaics . . . I General points: the photovoltaic effect 2 Initial attempts using organic materials: the phthalocyanines . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3 Solar cells based on pentacene doped with iodine . 4 The general principle of Graetzel and current organic solar cells II n-Conjugated materials under development for the conversion of solar energy I Metal-Insulator-Metal structures 2 How bilayer hetero-structures work and their limits 3 Volume heterojunctions . . . . . . . . . . . . . . . . . . . . . .. III Additional informations about photovoltaic cells and organic components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I Discussion about mechanisms leading to the generation of charge carriers in organics . . . . . . . .. 2 Electric circuit based on an irradiated pn-junction ; photovoltaic parameters 3 Circuit equivalent to a solar cell . . . . . . . . . . . . . . . . 4 Possible limits 5 Examples ; routes under study and the role of various parameters 6 Conclusion .. .. . ... .. . . . . . . . .... . .. . . . . . .. . .

288 290 292 292 294 294 294 296 296 298 304 306 309 310 313 313 313 316 318 320 321 321 322 325 328 328 330 334 336 337 339

xxvi Contents Chapter XII: The origin of non-linear optical properties . . . . . . . . . . . . . . . I Introduction: basic equations for electro-optical effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. I Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2 Basic equations used in non-linear optics II The principle of phase modulators and organic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Phase modulator 2 The advantages of organic materials 3 Examples of organic donor-acceptor non-linear optical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 General structure of molecules used in non-linear optics III The molecular optical diode . . . . . . . . . . . . . . . . . . . . . . . . 1 The centrosymmetric molecule 2 Non-centrosyrnmetric molecules . . . . . . . . . . . . . .. 3 Conclusion IV Phenomenological study of the Pockels effect in donor-spacer-acceptor systems I Basic configuration 2 Fundamental equation for a dynamic system . . . . . 3 Expressions for polarisability and susceptibility . . 4 Expression for the indice-and the insertion of the electro-optical coefficient r . . . . . . . . . . . . . . . . . V Organic electro-optical modulators and their basic design I The principal types of electro-optical modulators . 2 Figures of merit . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3 The various organic systems available for use in electro-optical modulators VI Techniques such as etching and polyimide polymer structural characteristics . . . . . . . . . . . . . . . . . . . . . . . . . .. I Paired materials: polyimide/DRI 2 Device dimensions-resorting to lithography . . . .. 3 Etching .. ..... ........ .. .. . .. .. .. .. .. . .. ... 4 Examples of polymer based structures VII Conclusion.. .. . .. .. . . . . . . .. . . .. . .... .. . . ... . . . . .

341 341 341 341 343 343 345 346 348 349 349 350 351 353 353 355 355 356 358 358 359 361 363 363 364 365 367 368

Appendices Appendix A-I: Atomic and molecular orbitals I Atomic and molecular orbitals. . . . . . . . . . . . . . . . . . . . . . 1 Atomic s- and p-orbitals 2 Molecular orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . .

373 373 373 376

Contents

XXVII

3 a-and n-bonds The covalent bond and its hybridisation I Hybridisation of atomic orbitals . . . . . . . . . . . . . . .. 2 sp3 Hybridisation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

380 381 381 383

Appendix A-2: Representation of states in a chain of atoms . . . . . . . . . . . . . . A chain of atoms exhibiting a-orbital overlapping I a-orbitals and a compliment to the example of 8 atoms in a chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 General representation of states in a chain of overlapping as-orbitals. . . . . . . . . . . . . . . . . . . . . . . 3 General representation of states in a chain of overlapping a p-orbitals II n Type overlapping of p-orbitals in a chain of atoms: n-p- and n "-p-orbitals III a-s- and o-p-bonds in chains of atoms . . . . . . . . . . . . . . . . IV Comments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I The Bloch function 2 Expression for the effective mass (rn")

389 389

II

Appendix A-3: Electronic and optical properties of fullerene-C60 in the solid (film) state I Electronic properties of fullerene-C60 II Optical properties and observed transitions Appendix A-4: General theory of conductivity for a regular lattice I Electron transport effected by an external force and its study I Effect of force on electron movement and reasoning within reciprocal space 2 Boltzmann 's transport equation II State density function, carrier flux and current density in the reciprocal space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 General expressions for fluxes of particles . . . . . .. 2 Expressions for the state density function. . . . . . .. 3 Expression for flux 4 Expression for current density in reciprocal space III Different expressions for the current density I Usual expression for current density in energy space 2 Studies using various examples 3 Expressions for mobility. . . . . . . . . . . . . . . . . . . . . . 4 The Kubo - Greenwood expression for conductivity

389 391 393 393 394 395 395 396

397 397 40 I 403 403 403 404 406 406 406 408 408 409 409 410 412 413

xxviii Contents IV

Complementary comments. . . . . . . . . . . . . . . . . . . . . . . .. 414 1 Concerning the approximation of the effective mass and isotropic diffusions . . . . . . . . . . . . . . . . .. 414 2 General laws for changes in mobility with temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

Appendix A-S: General theory of conductivity in localised states . . . . . . . . . I Expression for current intensity associated with hopping transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1 Transcribing transport phenomena into equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2 Calculating the current intensity due to hopping mechanisms II Expression for current density and thermally activated mobility 1 Expression for current density relative to transport at a particular energy level 2 Generalisation of the form of Kubo-Greenwood conductivity 3 Thermally activated mobility III Approximations for localised and degenerate states Appendix A-6: Expressions for thermoelectric power in solids: conducting polymers I Definition and reasons for use 1 Definition . .. .. . . . . . . . ......... .. . . . . . .. . . . . 2 Reasons for use II TEP of metals (EF within a band of delocalised states) III TEP of semiconductors (SC) (EF in the gap) 1 Preliminary remark . . . . . . . . . . . . . . . . . . . . . . . . .. 2 An ideal n-type semiconductor . . . . . . . . . . . . . . . . . 3 An ideal n-type semiconductor . . . . . . . . . . . . . . . .. 4 Comment on amorphous semi-conductors 5 A non-ideal amorphous semiconductor with EF below its states in the band tails . . . . . . . . . . . . . . . . IV TEP under a polaronic regime . . . . . . . . . . . . . . . . . . . . .. 1 High temperature regime . . . . . . . . . . . . . . . . . . . . .. 2 Intermediate temperature regime . . . . . . . . . . . . . . . 3 Other regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V The TEP for a high density of localised states around EF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Initial hypothesis .. . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The result in VRH . . . . . . . . . . . . . . . . . . . . . . . . . . .

417 417 417 419 419 419 420 420 421

423 423 423 423 424 424 425 425 426 426 426 427 427 427 427 427 427 428

Contents xxix VI VII

General representation Real behaviour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 General laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2 Behaviour as a function of doping levels . . . . . . . .. 3 Representational graph . . . . . . . . . . . . . . . . . . . . . . . 4 An example result

AppendixA-7: Stages leading to emission and injection laws at interfaces I Thermoelectric emission and the DushmanRichardson law II Schottky injection (field effect emissions) 1 The potential barrier at the atomic scale . . . . . . . . . 2 Emission conditions : Schottky emission law and the decrease in the potential barrier by field effect . III Injection through tunnelling effect and the Fowler-Nordheim equation 1 The problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2 Form of the transparency (T) of a triangular barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Fowler-Nordheim equation. . . . . . . . . . . . . . . . Appendix A·8: Energy levels and permitted transitions (and selection rules) in isolated atoms I Spherical atoms with an external electron . . . . . . . . . . . .. 1 Energy levels and electron configuration . . . . . . . .. 2 Selection rules II An atom with more than one peripheral electron . . . . . . . I First effect produced from the perturbation Hee due to exact electronic interactions 2 Perturbation involving the coupling energy between different magnetic moments exactly tied to kinetic moments. . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Selection rules Appendix A-9: Etching polymers with ion beams: characteristics and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. I Level of pulverisation (Y) . . . . . . . . . . . . . . . . . . . . . . . . .. I Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2 The result Yphysical = f (E) : 3 zones 3 Level of chemical pulverisation . . . . . . . . . . . . . . .. II The relationship between etching speed and degree of pulverisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1 At normal incidence . . . . . . . . . . . . . . . . . . . . . . . . . . 2 At oblique incidence

429 429 429 430 431 431

433 433 434 435 435 437 437 438 440

443 443 443 444 445 445

446 447

449 449 449 450 451 451 451 452

xxx Contents III IV

V

Speed of reactive etching (IBAE Ar +/02 or 0 +/ 02) . .. Preliminary modelling of Yphysical for PI 2566. . . . . . . . . 1 Levels of carbon pulverisation using 0 + ions ... . 2 Comparing simulations of Yphysical (6) = f (6) and the Thomp son and Sigmund models . . . . . . . . . . .. Results from etching of polyimide s . . . . . . . . . . . . . . . . . . 1 Self-supporting polyimide: UPILE X . . . . . . . . . . .. 2 A study of the etching of PI 2566 "

AppendixA-I0: An aide-memoire on dielectrics I Definitions of various dielectri c permitti vities . . . . . . . . . 1 Absolute permittivity. . . . . . . . . . . . . . . . . . . . . . . .. 2 Relative permittivity 3 Comple x relative permittivity . . . . . . . . . . . . . . . . .. 4 Limited permittivities 5 Dielectric conductivity 6 Classification of diverse dielectric phenomena II Relaxati on of a charge occupying two position separated by a potential barrier 1 Aide-memoire.. . . . .. . . . . . . . . . . .. . . . . . . . . .. . . 2 Transportation in a dielectric with trappin g levels, and the effect of an electric field on transition s between trap levels . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Expression for the polari sation at an instant t following the displacement of electrons 4 Practical determination of potential well depth s. . . Appendix A-ll: The principal small molecules and polymers used in organic optoelectronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I Chemic al groups and electron transport II Examples of polymers used for their elect roluminescence I The principal emitting polymers 2 'The' polymer for hole injection layers (HIL) 3 Example of a polymer used in hole transport layers (HTL) 4 Example of a polymer used in electron tran sport layer (ETL) III Small molecul es 1 The principal green light emittin g ligands 2 Principal electron transporting small molecules emitt ing green light 3 Example electron transporting small molecules emitting blue light 4 Example small molecules which emit red light

452 454 454 454 455 455 456 459 459 459 459 460 460 461 461 463

463

464 466 467

471 471 471 471 472 473 473 473 473 474 474 474

Contents 5 6 7

XXXI

Examples of small molecules which serve principally as hole injection layers (HIL) 475 Examples of small molecules serving principally in hole transport layers (HTL) 475 Example of a small molecule serving principally to confine holes in 'hole blocking layers ' (HBL) .. 476

Appendix A-12: Mechanical generation of the second harmonic and the Pockels effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 477 I

II

Mechanical generation of the second harmonic (in one-dimension) I Preliminary remark : the effect of an intense optical field (E W ) • • • •• • • • ••• • • • • • • • • • • • • • • • •• 2 Placing the problem into equations 3 Solving the problem Excitation using two pulses and the Pockels effect. . .. I Excitation from two pulses 2 The Pockels Effect . . . . . . . . . . . . . . . . . . . . . . . . . ..

477 477 477 480 481 481 482

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 485 Index

495

I

Band and electronic structures in regular I-dimensional media

I An introduction to approximations of weak and strong bonds Historically, the relatively simple theory of free electrons moving within the confines of a flat-bottomed well, the walls coinc iding with physical limits, has been used to understand the conduction of electrons in metals. Classically, in a 3 dimen sional system a potenti al box is used as detailed in [kit 96] or [ash 76]. To more fully describe the electrical properties of metals, semiconductors and insulators, however, increasingly elaborate model s are required which take into account precise interactions between electrons and their environments. Depending on the nature of the bond ing in the solid, two different approaches may be used .

1 Materials with weak bonds The potential box model can be refined by imposing on electrons a potential generated by a regularly spaced crystal lattice , in which the Coulombic potential varies inversely to the radius (l /r) of each electron from an ion at any lattice node . In Figure I-I , the row of atoms along axis Ox with periodic spacing a, the lattice constant , has electrons in an orbital s radiu s R (Figure I-I-a). A I-dimensional representation of the potential energy of electrons is given in Figure I-I-b, with the cond ition that a < 2R. Admittedly, in using the term ' potential energy ' rather than ' potential' , we fall into a slight linguistic error, however the term 'potential energy ' is widel y used in material physics and quantum mechanics. When an electron moves towards any nucleu s in the lattice, the potential it undergoe s varies according to the direction defined by line Ox, which runs through the nuclei . Changes in potential with respect to Ox have no actual physical meaning as conducting electrons are actually situated on the outer layers of atoms. In using line D, however, which does not pass through lattice nuclei, the electron to nucleus distance no longer tends to zero and potential s going to finite values can be resolved . In addition, with the superposition of two potential curves , due to a < 2R, the barrier

Optoelectronics of molecules and polymers

4

(a)

x

Potential Energy (b)

Figure I-I. Weak bonds: (a) atomic orbitals (s type with radius R) in a regular latticeof periodicity a obeying the condition a < 2R; and (b) l-dimensional representation of the resulting potential energy (thickline) observedby electrons.

which exists mid-way between two adjacent nods is lowered, and thus, for a solid , a potential with periodic fluctuation s can be resolved. The initial repre sentation using a flat-bottomed well-which implied that electrons are free electrons-is now replaced by a wavy-bottomed well and to a first approximation in l -dimen sion (i.e. r == x), the potential is now defined by V(x) = WQ cos 2: x. As a decreases below 2R, WQ also decreases and with it perturb ations due to the crystal lattice. In reality, treatment of this problem by the method of perturbation becomes increasingly plausible as the value of a decreases with respect to 2R. An approximation for a half-free electron, according to the Hamiltonian 151 order of approximation, H(l) = WQ cos 2arr x, is an improvement upon that of a free electron (which ignores H(l» . As an electron delocalised within a metallic lattice has a low value of WQ, the theory developed for weak bond s can also be applied to bonds within metals.

2 Materials with strong bonds The way in which strong bond s are treated closely resembles a chemi st' s point of view, as the properties of a solid are derived from orbitals of constituent atoms and the chemi cal bonding can be specified using a linear combination of atomic orbitals. This reasoning is moreover acceptable when the electrons under consideration are attached closely to bonding atoms, as approximations of strong bonds are justifiable when a > 2R is obeyed (Figure 1-2-a). An approximation can therefore be made for covalent solids in which valence electrons ensuring bondin g are localised between two atoms (for bondin g states in more detail, see Appendix A). However, if we study potential curves using Ox, we obtain a function which diverges when electrons are close to nuclei. While discontinuity

I Band and electronic structures in regular I-dimensional media

5

(a)

x

Potential Energy (b)

r

Potential from D for a ~2r l.e. strong bond

Deep, independent potential well, degenerated N times, at level E in which a» 2R loc

Figure 1-2.Strong bonds: (a) atomic orbitals (s type with radius R) in a regularlattice (lattice constanta) obeyingthe condition a ~ 2R; and (b) I-dimensional representationof the resulting potential energy (thick line) observed by electrons.

is removed when line D is used relative to valence electron s, two scenarios may be envisaged: -

-

If a » 2R, very deep potential wells appear because there is practically no overlap between potentials generated by adjacent nuclei. Taking this argument to its limit, if a chain of N atoms and N valence electrons is formed with sufficiently long bonds to make N electrons independent in N deep , independent potential wells, the energy levels (Eloc in Figure 1-2) are degenerated N times, are all identical and therefore indiscernible from one another. If a ~ 2R, the decreasing gap between atoms induces a slight overlap of potentials , generated by adjacent nuclei, and potential wells, therefore, are no longer independent. Subsequently, degeneration occurs in which the electrons of one bond interact with those of neighbouring bonds to break-up energy level bands . It should be noted though that the potential well s are, however, con siderably deeper than those found for weak bonds (a < 2R), so much so that electrons rema in considerably more in the locality of each atom to which they are attached. We should also note that , unlike for weak bonds, simple treatment using the perturbation method is no longer possible due to the con siderable depth of the wells.

Most of the remainder of this chapter will detail the determination of energy levels and bands in I-D for strong bonds . Beforehand though, and by way of comparison, we will con sider the origin of energy band s due to weak bonds .

6

Optoelectronics of molecule s and polymers

II Band Structure in weak bonds 1 Prior result for zero order approximation While the description of weak bond s corresponds essentially to that of metallic bond s, it would be more appropriate to outline reasons for the origin of potential band s in which electrons are found , as represented in Figure 1-3 which follow s on from Figure I-I . Initiall y we should remember that for free electrons (zero order approximation) potentials correspond to that of a flat-bottomed well (horizontal line joining nodes in Figure 1-3). For this system where the potential (or potential energy ) V = VO = 0, the amplitude in Schrodinger's wave equation is expre ssed as ~\jIo + ~E\jIo = 0. By favouring real solutions and assuring the propagation of a wave assoc iated with an electron by using k 2 = ~~ E, we have \jIo = Ae± ikX; elsewhere, the curve E = f(k) is obtained from E = EO = ;~ k 2 We have seen that lattice effects result in a perturbed periodic potential. From this the first term of a developing Fourier serie s allows an approximation to be made:

V :::::: V(l) = w(x) =

W Q cos

2n -x. a

The function of the wave is itself perturbed and takes on the form of a Bloch function i.e. \jIk (x) = \jIou(x) = eikxu(x), in which u(x) is a periodic function based on the lattice constant a. The reason for this is that the wavefunction must remain unchanged while undergoing translations of modulus a (Ta ) which imposes u(x) = u(x + a): cf. rem arks in Appendix A-2, Part IV.

2 Physical origin of forbidden bands Figure 1-4 shows a regular chain of atom s and two incident rays. Incident rays are reflected by lattice atom s and co nstructive interference occurs when the difference in step ( ~) between two waves is equal to the ray wavelength multiplied by a whole number. In a 1-D system, the differ ence ~ of parallel waves 1 and 2 after reflection is ~ = 2a, however, incident waves undergoing maximum reflection will have wavelength A. n such that ~ = 2a = n A. n . In addition, we should note that if the vector modulus Potential energy ( patial origin taken from lattice node). w(x) =

WQCO

2lt x a

a

o Figure 1-3. Potential energy curve w(x)

x

= Wo cos 2an x showing wo <

O.

Band and electronic struct ures in regular I-dimensional media

a

7

:

Figure 1-4. Bragg reflection for a I-D .

of a wave is k = 2,n , incident waves with k = k n = ~n = n ~a undergo the maximum r-.. I~n reflection , according to Bragg's condition. Usi ng the zero order approx imation, the amplitude of an incident wave assoc iated with an elec tron in a weak bond is only slightly perturbed such that 1jJ~ = Ae ikx. The time dependent incident wave is therefo re written as [\IJ~ (x, t) line. = Aei(kx- wt) and is expressed as an incident wave pro gressing in the direction x > O. When k = k n exactl y, the reflected wave tends towards x < 0 and [lIt (x, t)]refl . = Aei(knx+wt). When stationary waves are formed from the superpos ition of incident and both symmetric and asy mmetric solureflected waves, for which k = kn = n tions are formed i.e. \IJ+ n cos( nan x)e -iwt and \IJ- n sine nan x)e - iwt, respectively. Concerning electrons fulfillin g k = kn = n for these two wavefunctions there are two corresponding probable electro n densities: p+ = \IJ+ \IJ+*n cos( nan x) and p- = \IJ- \IJ- *n sin 2 (n; x). For any single given value of kn , the probable electro n densities p+ , p", and p for electrons respect ively desc ribed by stat ionary waves \IJ+ and \IJ- and the progressive wave [\lJkn(X, t)] = Aei(knx- wt ) are show n in Figure 1-5. Th e progressive wave correspo nds to p = constant, and when k = kn • ca n only exist by neglectin g any reflections due to the lattice with spatial period a. With the zero order approx imation, V = VO = 0 ; when V "I Vo . this type of wave ca n only exist when k "I k n . In Figure 1-5, we ca n see that:

i,

i

p+ displa ys the highest conce ntration of electrons close to nuclei in a configuration which corresponds to the lowest average energy w+ (the small distance between electron and nuclei results in a high modulus of negative Co ulombic energy) ;

p (x)

Figure 1-5. Electron densities p+ . p" and p associated. respective ly, with stationary waves 1jJ+, 1jJ- and a progressive wave.

8

Optoelectronics of molecules and polymers

p- shows a maximum concentration of electrons mid-way between nuclei in a configuration' associated with the highest energy w: (high average electronnucleus distances result in low Coulombic potential moduli); at p = constant, where there is an equal distribution of electrons associated with an intermediate electron -nucleus distance , the configuration corresponds to an intermediate energy, which approximates to that of a free electron .

Finally, two different energy values are generated by two physical solutions, \11+ and \11-, for the same value kn of k(k = kn = n and this dispersion, or degeneration, of energy values is represented in Figure 1-6, in which E = f(k) . The 'gap' EG = w- - w" is a forbidden band because, at the same value of k = kn , there is a discontinuous change from values of energy w+ to w- .

i ),

3 Simple estimation of the size of the forbidden band The wavefunctions \11+ and \11-, normalised for a segment L = Na, corresponding to a I-D chain ofN + I ~ N atoms, are such that:

=

or

\I1+(x)

or

\I1-(x) =

If If

cos (;x) ,

sin (;x).

k

Figure 1-6. Curve E

= f(k) . The

zero order approximation, corresponding to perturbation

potential w(x) = 0, results in energy EO = !i;~2 . Thus lattice-electron interaction is considered negligible, just as for a flat-bottomed well. Through waves associated with electrons at atoms in a regular chain ,the lattice results in two solutions for energy for each value k = kn = n ~ , with w+ < w" as detailed in Section 11-2.

1 Band andelectronic structures in regular l-dimensional media

9

The gap in energy, equal to EG = w- - w" = - (w+ - w- ) gives us:

=

2wo {L 2 ( 2TI ) -T 1 cos --;-x dx. 0

As Wo must remain negative, the potential energy curve goes through a minimum at the nuclei where the origin is placed, as shown in Figure 1-3, and we obtain a positive value for EG = - woo As EG = Iwol, stronger electron-lattice interactions, and an increasing Wo, result in a larger forbidden band.

III Floquet's theorem: wavefunctions for strong bonds 1 Form of the resulting potential Figure 1-7 represents an infinite chain of atoms spaced distance a apart with a > 2R. As the atoms are numbered . . . (0) , (1 ) , (2), . . . (s - I) and the origin is at the nucleus (0), the distance between the nucleus of atom s from the origin is rs = s.a. Using the layout shown above, the potential generated by atom (s) at a point in space located by f, or vs(f), is the same as that generated by atom (0) at point fl, similarly denoted VO( fl). In giving f s = sa, and using f s + f' = f, we obtain vs(f) = VO(fl) = VO(f - f s). In addition to the potential generated by atom (s), any electron placed at r will undergo a potential Vs-l (f) = vo(f - fs- I) resulting from neighbouring atoms (s - I ) , and so on for other atoms. We can therefore sum the resulting potential in terms of f over an infinite chain: s=+oo V(f )

=

L

s=+oo vs(f )

=

s=-oo

'--

L

vo(f - f s).

(1 )

s=-oo

r.=sa Figure1-7. Atoms ina I-Dinfinite chain.

•

10

Optoelectronics of molecules and polymers

We can now note that the result ing potential, VCr) , is periodic with respect to a, and that symmetry imposed by an infinite chain results in a potential in r which is the same if calculated using r - a, . . . , r - sa, . .. and so on. We can therefore write, independent of s being positive or negative

(2)

VCr) = VCr - a) = vcr - sa) .

As the potenti al VCr), in which is placed each electron, is a periodic function, the corresponding wavefunction is a Bloch function . We shall use these results to establ ish Floquet's theorem in Section 3.

2 The form of the wavefunction As shown in Figure 1-8, if a valence electron on a nucleus s is not influenced by the presence of neighbouring nuclei, as the atoms are far apart, then the potential at an electron at r is reduced to vscr) = vocr - rs), and therefore VCr) ~ vo(r - rs). Subsequently, the obtained wavefunction corre sponds to that of a lone atom s with atomic wavefunction Wscr). If the atoms are moved close enough together, an electron placed in r will be influenced by more than one and its wavefunction corre sponds to a molecular orbital derived using the LCAO method (see Appendix A-I). That is: wcr) =

L csWs(r) ,

in which Wscr) is the wavefunction of an electron from atom s at r. We can thus see that the form ofthe solution chosen for a wavefunction is identical to that followed by the potential in which an electron is placed . By the reasoning developed above for an electron and the potential to which it is subject, the wavefunction Ws(r) relative to an electron positioned at r from an atom s is identical to that of the same electron placed at f' on an atom (0): Wscr) = wocr '),

or with r = r s + r ' , Wscr) = wocr - f s).

We can thus write , in a manner similar to that for the potential:

+00 w(r)

=

L

(3)

cswocr - f s)

s= -oo

(s -1)

(s)

(s+l)

Figure 1-8. Systemof effectively independent atoms.

I Band and electronic structures in regular l -dirnension al media

II

3 Floquet's theorem: effectof potential periodicity on wavefunction form For an electron at r belonging to an infinite, regular chain, the wavefunction can be written: -

in the form of a linear combination, +00

L

tV(f)=

(3)

cstVo(r-rs),

s= -oo

corresponding to Hiickel's wavefunction representation; or in the form of a Bloch function, under which the electron is subject to a periodic potential (see Part II-I), and

-

tVdr) = eikru(r) with u(r) = u(r + a).

(4)

Initially, the Bloch form applied to a wavefunction calculated in r + a results in: tVk(f + a) = eik(i'+a)u(r + a) = eikaeikfu(f) = eikatVk(r)

(5)

Using Huckel's expression (3) to develop tVk (r) as described in eqn (5), we obtain a first expression for tVk (r + a): tVk (r + a) = eikalcotVo(r) + Cl tVo(r - a) + . . . + c, tVo(f - sa)] .

(6)

Directly performing a Hilckel type representation of l/Jdr + a) yields: tVdr + a) = cotVo(r+ a) + cll/JO(f + a - a) + .. . + c, l/Jo(r + a - sa) +cs+'l/Jo(r+a-[s+lja)+ · ·· .

(7)

If we identify term by term through (6) and (7) we obtain: eikiico'1Jo(f) = cj '1Jo(f),

or

CI

= coe ik.a

eikaCltVo(f - a) = C2tVo
or C2 = cle ika

- = cs+, l/Jo( -r + -a - [s + l]a), eikacsl/Jo( -r - sa)]

or Cs+I=Cseika

We therefore have, as a general rule:

(8) We are thus brought to the final form of the wavefunction, or Floquet's theorem, which can be written in two equivalent ways (using the notation l/J(r) == l/Jk(f» : l/J(f) =

L csl/Js(r), with c, = Co eik'fs, l/J(r) = l/Jdr) = co L eik'f'l/Js(f);

(9)

or: +00

L

s=-oo

+00 CstVo(f - rs) = Co

L

s= -oo

eik'fsl/Jo(f - r s), with rs = sa.

(10)

12

Optoelectronics of molecules and polymers

IV A study on energy 1 Defining equations (with x == r: 1 - D) We shall consider a specific atom s and its neighbours. The resulting potential, around this atom, is V(x) as described by the dotted curve in Figure 1-9. Alternatively, if the atoms are sufficiently spaced out to be considered independent, the equivalent potential would be Uo(x). Figure 1-9 represents Uo(x), generated in terms of x by atom s, which is the same potential Uo(x + a) generated by atom (s + 1) at (x + a). This results in Uo(x) = Uo(x + a) , in which Uo is periodic due to there being isolated atoms in the chain. Equally, we can see that for x EJr s - ~,rs + ~ Lwe have V(x) ~ Uo(x). When W(x) = V(x) - Uo(x), for rs - ~ < x < r, + ~ ,W(x) is small ; in addition, as Vex) < Uo(x) and now W(x) < o. Detailing the Schrodinger wave equation: -

for an electron described by the wavefunction \11k (x) and placed in the resulting potential V(x), in which \11k (x) and Vex) take into account effects from neighbouring atoms : (11)

-

or for an electron belonging to an isolated atom s: the wavefunction of an electron positioned in terms of x from the isolated atom is \I1s(x) = \I1o(x - sa), while the

Potenti al Energy f .-I

•

r . -al2

,

f.

l

'.

\\

\.

'........ Figure 1-9. The function W(x)

= V (x) -

Uo(x).

x

•

I Band and electronicstructures in regular I-dimensional media

13

potential is Uo(x) = Uo(x - sa). Noting that its energy is Eo, we now have Eo'1Jo(x - sa)

tt2

= - 2m II'1Jo (x -

sa) + Uo(x)'1Jo(x - sa).

(12)

Multiplying the two parts of eqn (12) by coeikrs (with rs = sa), and integrating over s:

Eo

~ co'''''$o(x - '.) ~ - ;:" [ ~ co';"'$o(x - ,.)] + Uo(x) L eikrs'1Jo(x - r.) .

Taking into account eqn (10), this can be rewritten as: Eo\IJk(X)

tt2

= --ll'1Jdx) + UO(X)'1Jk(X). 2m

(13)

In taking the difference [eqn (11) - eqn (13)], we obtain: (E - Eo)\lJdx)

= [V(x) -

(14)

UO(X)]\lJk(X)

And the potential, V(x), if periodic, gives Vex) = V(x - rs) and thus: W(x) = [V(x) - Uo(x)] = [V(x - rs )

-

Uo(x - rs)1 = W(x - r.) .

(15)

We should note that W(x - rs) is a function with period a, and is consequently independent of s even though the parameter can be brought into or taken out of the integral sum. Developing eqn (14) gives \lJk(X) according to Floquet's eqn (10): (16)

2 Calculation of energy for a chain of N atoms We can directly obtain the following equation by multiplying eqn (16) by \IJ~(x) and integrating over a range of N atoms from 0 to (N - l) : (E - Eo)('1Jdx)l'1Jk (xj) = Co

L eikr, f '1J~ (x)W(x - rs)'1Jo(x - r

s)

dx.

s

With eqn (10), we obtain s=N -l t=N -l

(\lJk(x)I\lJk(X» ) = Icol2

L L s= O

(=0

e i(ks-kt)a('1Jo(x - rs)I'1Jo(x - rd) ·

(17)

14

Optoelectronics of molecules and polymers

As ('!ro(x-rs)I'!ro(x - rl») = 8~ (with 8~ = 1 if s = t, 8~ = 0 if s s=N- l I=N-l

L L

(WkCx)IWk(X» ) = Icol2

s=o

ei(ks-kl)a 8~ = Icol2

1=0

s=N- l

L

f= t),

ei(ks-ks)a = Ico1 2.N

s=o

Normalisation of the function Wk(X) : (WkCx)IWkCx»)

= 1 = Icol2 . N, gives

1 Co = .)N'

(18)

From eqn (17) we can deduce that with Wk(x) = c~ LI e-ikrt'!r;(x-rd, (E - Eo) = Icol2

L eikrs L e-ikrt f '!r~(x - rl)W(x - rs)'!ro(x - rs) dx, s

I

and through eqn (18), E-

Eo = ~

L eik(s-I)a f '!r~(x -

ta)W(x - sa)'!ro(x - sa) dx.

s.t

The latter can also be written: E = Eo + ~

L eik(s-I)a('!ro(x - ta)!W(x - sa)I'!ro(x - sai ).

(19)

s.t

Using Huckel 's approximations, which assume couplings only with first neighbours: ('!ro(x - ta)IW(x - sa)I'!ro(x - saj ) = ('!rtlWI'!rs)

=

I

- a for s = t(with a > 0 as W < 0) -~ for s = t ± I ofor all others. (20)

(As W < 0, we also have ~ > 0, if orbitals WI and \lis are of the same sign: cf. Appendix I, Section IV). Given the approximation, the integration of energy in eqn (19) over s from So = 0 to SN-l = N - I) yields: E=

Eo +

k[L

eik(sO -I)a('!rtlWI'!rso ) +

L

I

+... + ~ ,'k('N-. -<), (o/dWlo/'N_' )]

eik(SI-I)a('!rtlWI'!rst)

I

(21)

I Band and electronic structures in regular I-dimensional media

15

Each term between the square brackets has N terms, as s takes on N values, and therefore each term yields the same contribution, in the form: "" eik(Sj -tl a(1/JtlWI1/Js·) = eik(O)a( -a) + eik(-l)a(-B) + eik(+l)a( -~) L..J ) ~ '-v-' '-v-' t

Sj=t ~1

Sj= t

Sj = t+ 1

= -a - Be -vika - Be ika Finally, the term inside the square brackets of eqn (21) is equal to N[ -a -

~e-ika - Be ika], and thus E can be expressed as:

. - ~elka . = Eo - [ a E = Eo - a - Be-Ika = Eo - [a

ika

+ 2~ (e +2e-

ika)]

+ 2~coskal.

This can be rewritten as: (22)

I E = Eo - a - 2B cos ka I

Figure 1-10shows a graphical representation of the energy dispersion curve E = f(k) . The amplitude of variation in E, with respect to k, is equal to 4B. Appropriately, allowed bands increase in size with higher values of ~ due to strong interactions between electrons on neighbouring atoms.

..

~J. Eo

···r················································

................................................... Eo-a-2p

.

~

k

- tu«

- 1t/2a

o

1t/2a

n/a

Figure 1-10. Energy dispersion curve for E = f(k) , within Hiickel and strong bond approximations.

16

Optoelectronics of molecules andpolymers

3 Additional comments: physical significance of terms (Eo - a) and B; simple calculation of E; and the appearance of allowed and forbidden bands in strong bonds. a Physical significance of terms (Eo - a) and

~

From eqn (12), Eo lJIo(x - sa) = - ;~ ~ '!Jo(x - sa) + Uo(x)'!Jo(x - sa), and with eqn (20) we obtain:

h2 Eo = ('!Jo(x - sa)l- -~ + Uo(x)I'!Jo(x - sa»)(12') 2m -a = ('!Jo(x -sa) IW(x-sa) I'!Fo (x-sa») (one of Huckel conditions in eqn (20»

I

so with W(x - sa)

= W(x) (eqn (15) gives periodicity W):

Eo - a = ('!Jo(X - saj] - : : .

2

= ('!Js(X) I - :m

(Eo - a) = ('!Js(X) I - : :

~ + Uo(x) + W(x) I'!Jo (x -

sa»)

~ + Uo(x) + W(X)I'!Js(x») , or:

~ + V(X)I'!Js(X»),

as V(x) = Uo(x) + W(x).

(23)

From the terms within brackets we can now say that: -

Eo represents the energy of an electron situated on a given atom s subject to a potential, Uo(x), resulting from the given atom; and -a represents the energy of an electron situated on a given atom s subject to a potential, W(x), resulting from that given atom and its neighbours; the term (Eo - a) = E~ represents the energy of an electron situated on a given atom s subject to a general potential, Uo(x)and W(x), resulting, respectively,from that given atom and its neighbours, as shown in Figure 1-11.

From eqn (20) we have seen that -~ = ('!Jo(x - ta)IW(x - sa)I'!Jo(x - sal) = ('!Jt IWI '!Js), with s = t ± 1. This term corresponds to the coupling energy of an electron, at given atom s, with electrons at adjacent atoms (t) and such that t = s ± 1, which are in the same orbital. This coupling is brought about by the intermediate perturbation potential (Wtx) resulting from neighbouring atoms which gives rise to an increase in degeneration, shown in zone II of Figure 1-11, corresponding to the allowed energy bands in Figure 1-10. As lattice atoms are brought together and a decreases the perturbation W increases and the permitted band increases in size as shown in zone II of Figure 1-12. Elsewhere [Ngu 94], the term -~ can be seen as the energy of the electron population associated with the covering integral ('!Jo(x - ta)I'!Fo(x - saj) .

Band and electronic structures in regular I-dimensional media

17

decreasing E~ + 2~ 4~(a)

r---------------,

Degenerated energy levels of electrons in state IjI and belonging to neighbouring , independent atoms (without coupling) .

Increasing degeneration

~allowedband) E~ -2~

Coupling effect (~) between electrons of the same electronic state (1jI) belonging to adjacent atoms.

(I)

(II)

Figure1-11. Scheme of coupling effect between electrons in the same electroni state. In zone II, the interaction between atoms increases, and a decreases, with ~(a) .

Gap between atoms (a decreases) (Allowed band)

(Allowed band)

No coupling of 2 degenerated levels, corresponding to states IjI and lp of each atom with independent neighbours .

Coupling effect of adjacent atoms each with two, different electronic states IjI and lp (for example J3~ =(lptIW!lp,»).

Figure1-12. Scheme to first approximation of formation of a forbidden band from electrons with different states of'" and

Ip.

b Comment on simplified energy calculation With H = -

i~!::J. + Vex) we

have following eqn (23)

(Eo - (1) = Eb = (\fs(x)IHI\fs(x)). Often the term (Eo - (1) = Eb is noted as -11 as it can be obtained by a more direct calculation, which inconveniently also hides certain physical realities.

18

Optoelectronics of molecules andpolymers Thus, the following can be written:

L L eik(S-t)a(wsIHIWt) E = E(k) = (Wk IHlwk) = _sc=--=--t _ (WkIWk) L L eik(s-t)a (WsIWt) . s

t

By using the relationship (WsIWt) = Sst and considering that Hiickel approximations can be given as Hss = (WsIHIWs ) = -a = Coulomb integral = negativeconstant, given the origin to be the energies of an electron at infinity (cf. p. 58 of [Ngu 54]); Hst = (WsIHIWt) = -~ for s # t with s and t as adjacent neighbours (-~ < 0, is the resonance integral between electrons sand t); and as the double integration is equivalent it can be ignored, saving a not inconsiderable amount of time and tedium, we can directly derive E = E(k)

=

-(X -

~eika - ~e -ika

=

-(X -

2~ cos(k.a) .

In this latest Hiickel approximation: -

first, the term Hss = -(X is identical to the term E~ = Eo - (X of Part IV-2 second, the term Hst = -~ for s = t ± 1, can be written as: Hst = (WsIHIWt) = (ws

= (ws

1- ~!:J. +

1- :~!:J. + V(X)I Wt)

Uo(x) + W(X)I Wt)

= Eo(WsIWt ) + (wS/WIWt ) = (Ws IWIWt). in which the initial value of -~ is found, as proposed in Section IV-2.

c Conditions for the appearance of allowed and forbidden bands For a forbidden band to appear within a system (zone II of Figure 1-12), at least two distinct band levels (zone I of Figure 1-11), need to disintegrate. It is here that we realise that each atom in a chain presents two distinct orbitals, wand
I Band and electronic structures in regular l-dimensional media

19

V I-D crystal and the distorted chain 1 AB type crystal Taking a 1-D crystal formed of alternating A and B type atoms, separated by distance a, with a total of 2n = N A or B atoms as described in Figure 1-13 , we can easily see that the repeat unit is 2a. For electrons on atoms A and B, with corresponding wavefunctions
l'!rk ) = I)Vkjl
and by following Floquet's theorem, Vkj = ak exp(ik2ja) and Tlkj = bk exp(ik2ja) . In going further with the hypothesis that there is a slight superposition of adjacent neighbouring orbitals, (
and

(
and we obtain a wavefunction for which normalisation can be performed using lak 2 + Ibkl2 = 1): 1

N/2

l'!rk) = N-

12 /

L exp(ik2ja)[akl
j=!

The equation, in proper values, is HI'!rk) = Ekl'!rk), with h2

H=-2m~+ LVAj+ LVBj, j

j

in which the potentials VAj and VBj are defined for a lone atom . Multiplication of the last equation by I
o

B1

•

A2

0

~:~

82

•

o

AI

•

/I • //f----+l,.....-_---+lo

-f7-

8n

~O

Figure 1·13. Alternating chain of n pairs of A and B atom s, with n varying from I to 1'1 /2.

20

Optoelectronics of molecules and polymers E ..............................................."" ....

-1-~

: ...!

I ,""'-.....-"",

_._

_

j .

State 'fi t

Figure 1-14.Dispersion curve E = f (k) of diatomic system AB.

2 The distorted chain a Representation In contrast to an alternating chain , in a distorted chain each atom (A) is u = ± ¥ apart, resulting in the distribution shown in Figure 1-15. Whil e the repeat unit is 2a, as A and A ' are identical, EA = EN. Figure 1-16 details the bands obta ined. For comparison, the dashed line describes E(k) for an undistorted chain with repe at unit length a, while the continuous line describes E(k) for the distorted chain with repe at unit length 2a. The distorti on provokes a gap EG with k = ±n / 2a .

b Conditions required for calculating a distorted chain Only by considering the energies Edefor and Erel ax can it be seen if a chain will undergo deformation. The former term is the energy necessary to produ ce a deformation between two atoms, and the latter the relaxation energy gained following the opening of a band gap .

•

Aj_1

•

o Aj_1

o

Figure 1-15. Distorted chain of identical atoms A and A' separated by alternating short and long bonds.

Band and electronic structuresin regular I-dimensional media

_~-----L-----

i

:.

"

I

.

--- - - -1- - - - - - -...--+-

:

i ' \ iI

.

/'

.

: kE

il

\!

I

V

-rc/a

rc/a

:

i

21

k

I

i

~-------I-----

I

Figure 1-16. Dispersion curves E = f(k) for a distorted crystal (continuous line) and undistorted (dashedline). In supposing: ('PA' IHI'PA ) = -~l and ('PA IHI'PA ~ ) = -~2, we obtain J-I

J

J

J

[Ben 91]: ~ECk=lt /2a) = EG = 2(BI - ~2) ' In addition, supposing that ~l and B2 are of the form -~l = -~ + au and -~2 = -~ - au (with a > 0 [Caz 96]), we finally obtain Eg

-

~

4a lui.

Edefor - this is the energy due to changing from a structure with regular spac-

ing (a) to a structure with repetition unit of 2a, consisting of alternating long (a + at) and short links (a - aj ). Ed efor can be described by Edefor = !Nkc12u12 = 2keNu 2, in which N is the number of atoms of type A or A' and 12ul is the stretching or contraction modulus of the 'spring' which mechanically joins two atom s together. In addition, we can note that for two atoms harmonically coupled by an elastic force fe of con stant ke , fe = - k , (2u) ; if 2u > 0, fe < 0 and is therefore an attractive force . Erel ax corresponds to the reduction in the electronic energy of a system going from the filled energy band of a system with period a (of energy E) to that with period 2a (denoted E-). Figure 1-16 shows the approximate value of Erel ax, corresponding to the reduction in energy of the most energetic electrons which are susceptible to participating in conduction bands . Section VI-3 detail s a more rigorous determination of Er elax.

If L\E = Edefor - Erel ax < 0, i.e. the energy gained by relaxation is greater than that required for deformation, there is a reduction in energy and the system = O. Following is stable when the value of deformation, un, is such that [aCtEl ] U=Uo

dimerisation, or rather the generation of alternating bond lengths, a gap is formed with = ±1t /2a which is termed the Peierls metal -insulator transition .

22

Optoelectronics of molecules and polymers

VI Density function and its application, the metal insulator transition and calculation of Erelax 1 State density functions The density function of each state can be calculated with respect to the dimension of energy, or the dimension of the space reciprocal (k).

a Definition of state density functions The state density function within energy space, denoted as Z(E), is such that Z(E) dE represents the number of electronic states within a unit volume , each described by a wavefunction and having energy between E and E + dE. For a 1-0 system, L = 1. In the same manner, when considering a unit volume defined by k, the state density function is described by n(k) such that n(k) dk represents the number of

dk.

The corresponding electronic states for which the vector k is between k and k + Figure 1-17 shows that k, which can be negative or positive (privileged solutions for wavefunctions may equally propagate in negative or positive values of k), in 1-0, should give: Z(E) dE = 2n(k) dk Byway of explication, in terms of space energy the interval dE corresponds, in k space,

dk.

to an interval of 2.dk which is situated between k and k + This correspondence can be seen more clearly though using Figure I-17-b by reducing the 20 model to the 10 model; once the k, axis is considered only, we can see that there are two intervals dk (dotted line with k x ) .

b Determining state density functions In evaluating n(k) we note that in reciprocal space, electrons are divided between cells of size Ak = 2~ which corresponds to the quantification of space in k obtained

E

(a)

(b)

E +dE

i I

I+-k k +dk

dk

k

k +dk

Figure 1-17. Relationship between energy space E and space k with E and (b) 2D.

= n2 k 2 / 2m : (a) 10 ;

I Band and electronic structures in regular I-dimensional media

23

I

from the periodic limit condition (PLC) , or Born-Von Karman condition, which is such that for a network of length L = Na, '!JkCx) = '!JkCx + L). With '!Jk (x) = eikxu(x)

=~~+W 'k( + L) =e l x u(x+L)

as u(x) = u(x + a) = u(x + 2a) = .. . u(x + L), ·~ . ::::} e' = I, or KL = 2rm , with whole values of n.

The difference between two consecutive values of k obtained for Lln = I is thus Llk = 2~ as detailed in Figure I-18-a. This interval corresponds, on average, to the electronic function '!Jk (x) , in other terms an actual state. We should note, however, that there are two functions corre sponding to two values of k taken at the extremities of the interval, but each of these functions is shared with this adjacent interval, giving on average one function per Llk. Interestingly enough though, on taking electron spin (x) and "'k (x) for the same state k, over into account, which allows two functions an interval Llk, we can place two electronic states, i.e.: n(k). Ak = n(k) . (2~ )L=J =

"'t

2, ::::} n(k) =

*;

(for L

=1=

I , the number of orbital s is N(k) = L.n(k) = ~) .

2 I

From Z(E) dE = 2n(k) dk, we can deduce that Z(E) = -

dE '

n dk

When con sidering a linear chain of N atoms , we can use eqn (22) of Section IV: E = Eo - a -

2~

dE cos ka, or- = dk

2~a

I I sin ka , in which : Z(E) = - -. - . n~a Sill ka

With . 2(ka) sIll2(ka) = I - cos = I-

2

E-Eo+a

2~

(

)

,

we obtain:

E

(a)

E

-----------

-E;~~+~2~---i

•

........ ......l!.t.. I

~""" " " " " """" 'I""""" ""'"

!

I ' I i I

i:

-ni«

i

-It/2a

----------------------

!

Eo

! 4~ !

Eo- a

;

I

i

I

...

(b)

I

,--- -----------------_.

•. . . . . . . . . . . . . . . . .• ••••••••••. . .• . . .• •••• . .} •••••

Eo - 2 a - 2 ~

!

Llk = 2lt/L

i i:

o

I

,

I I

k

It/2a

Figure 1-18. (a) Dispersion curve E atoms.

-It/a

!N

N(E) = Z(E).L

lt~

= f(k) ; and (b) density curve of states for a chain of N

24

Optoelectronics of molecules and polymers

For a complete chain, or length L = Na, the number of orbitals is defined by

N(E)

N

= LZ(E) = Na Z(E) = -

.~

x ----;:::=====

)1- (E+ ~~- F,j)'

2N 1 =. n J(2~)2 - (E + ex - EO)2' a graphical representation is shown in Figure 1-18.

2 Filling up zones and the Peierls insulator-metal transition a Filling up an undistorted zone For an undistorted chain of N atoms of length L = Na and N electrons in a single quantum orbital, the number of unit cells of dimension 2n/L = 2n/Na that we can fit into a Brillouin zone, of size 2n / a, is 2~n!~a = N (situated between k = - n / a and k = n/a). However, on taking spin into account , 2 electrons can be placed into each cell and therefore, over the whole zone or chain we can place 2N electrons . As the system liberates only N electrons, the zone shown by the dashed line in Figure 1-16 is half full and electrons there can move easily : the system is metallic in nature .

b The insulator-metal transition When distortion is energetically favoured, as detailed by the curve E = f'(k) in Figure 1-16 which corresponds to a distorted crystal, the zone has the dimension ic]« (as it cannot go beyond k = -n/2a and k = n/2a). For the distributed N electrons there are only N/2 cells available. In other words, the zone has been completely filled, and because there are no free places electrons cannot move about, as detailed by the interior of the curve E- = f (k) in Figure 1-16. This change of state, from metallic to insulator, is known as the Peierls transition.

c Fermi level wave vector and the position of EF If each atom liberates a single electron into a particular state on a chain of N atoms separated by period a, we can determine the wave vector kf at absolute zero for electrons at the highest energy level, otherwise called the Fermi level, which divides filled from empty states. At absolute zero, the Fermi-Dirac splitting function F(E) = 1 while E < EF (and F(E) = 0 while E > EF) and the number of electrons (N) for a chain of atoms of a certain length (L) can be calculated by utilising the functions of orbital density Z(E)

1 Band and electronic structures in regular I-dimensional media

25

or n(k) which are such that:

r+

oo

N=

JEmin

N=

f

F(E)N(E) dE =

rEF JEmin

N(E) dE =

l

EF

Emin L.Z(E)

dE

+OO N(k)F(k) dk = .- kF: N(k) dk = f- +kkFFL.n(k) .dk

- 00

I dk = -.(2kF) Na = . : Na ., in which kF = -11: . -kF 11: 11: 2a Figure I-18-a shows that, for the undistorted chain, the energy EF, for kF = ~ is such that EF = Eo - ex. For a distorted chain , in which there are always N electrons freed by N atoms, kF = ~ is retained; the Fermi level and midway between occupied and empty levels at 0 K is thus situated in the middle of the gap EG, as shown in Figure 1-16.

3 Principle of the calculation of Erel ax for a distorted chain In Section V-2-b, it was shown how Erelax corresponded to the difference in energy of N electrons on to an undistorted chain (End) and the energy of the same electrons on a distorted chain (Ed) i.e. Erelax = End - Ed. Figure 1-16 shows the approximate estimation of Erel ax determined by realising that: -

-

the undistorted chain, as described by the dashed curve E = f(k) in Figure 1-16, electrons participating in conduction are the most energetic and are within EF for k = kF = ~ situated at the intersection of the energy and 0 k axes ; and for the distorted chain , the zone between - ~ and ~ is only just full so that the most energetic electrons, which participate in conduction, are at the highest point of the band indicated by E- , otherwise written EA - 2exA .

Erel ax as shown in Figure 1-16 therefore shows the difference in energy of conducting electrons in undistorted and deformed chains . In going further, a more rigorous estimation than Erelax = End - Ed can be made using electrons in both calculations (of End and Ed) which fill cells between -kF and +kF in k space. Thus :

+k E(k)N(k)dk = f +Jt/2aE(k) -dk, L · f- kF -rr./ 2a F

End =

11:

with E(k) = Eo - ex - 2~ cos ka(E(k) for undistorted chain). Again, the energy Ed of a distorted system is given by Ed = r~:!22aa E- (k) ~dk, in which E- (k) is the energy function of a distorted system , as traced in Figure 1-16. More detailed calculations can be found elsewhere [pei 55] .

26

Optoelectronics of molecules and polymers

VII Practical example: calculation of wavefunction energy levels, orbital density function and band filling for a regular chain of atoms Herethe exampleis of a chain in I-D (alongx) consistingofN atomsregularlyspaced by distance a and of length L, such that L = Na. Specifically, we will be looking at a closed or cyclic chain which has N = 8 (Figure 1-19). 1 Limits of variation in k

The domain in which k can vary can be obtained by periodic limit conditions(PLC), otherwiseknown as the Bom-Von Karmenconditions. These conditionsindicate that for a cyclic chain the probability of finding an electron at a co-ordinate point x is unique and does not depend on the number of times the electron has gone around the chain. This can be expressedas l\J(x) = l\J(x + L). Giventhat the chainis periodic,the wavefunction can be written as a Bloch function in the form l\Jdx) = eikxu(x) with u(x) such that u(x) = u(x + L). The PLCs applied to the Bloch functionbrings us, as shownin PartVI-I -b, to eikL = I.As I = ei2pll and L = Na, we nowhave: k = kp = 2n (withp equalto zeroor a positiveor negative integers: p = 0, ± I, ±2, ±3 , .. .). Note that by using eqn (22), E = Eo - a - 2~ cos ka = Eo - a - 2~ cos kpa. The representation can be restrictedto a singleperiodas the curveE(k) is periodic. Identicalsolutionsfor energywouldresult from otherperiods.Weobtainthe so-called reduced zones for E (k has period such that -n < ka < n: with k = kp, -2} .::: kp .::: 2} (a reduction of Brillouin's first zone).As kp = 2n -~ p .::: ~ as N takes on successive integer values.

Ja

Ja'

.: :

2 Representation of energy and the orbital density function using N

Ja

=8

We havejust seen that the limits in variation of kp = 2n can be reduced to - 2} .::: kp '::: i · As N = 8 and kp = i1P, the successive values of kp (with -~ .::: p .::: ~,or p E [-4, -3 , .. . 0, . . . 3, 4]) are thus: n a'

3n 4a '

n 2a '

p-

°- - - -

n n n 3n n 4a ' ' 4a ' Za' 4a ' a .

k - -- - - -- --

....-.

8=0 7

6

3

5 Figure 1-19.

4

Representation ofa cyclic chain with N = 8.

Band and electronic structures in regular I-dimensional media E

E

I -----

27

(b)

J._= _ 2

2

------- I

o n ....1L 4a 4a 2a k_1 ko k l k1

_K _ 3n _....1L _....1L

a 4a 2a k-4 k_3 k_1

3n 4a k3

Figure 1-20.Representation for N = 8 of: (a) E

t

Number of states (c)

Z(E)

= f(kp); (b) Z(E ); and (c) number of states.

The expre ssion for energy from eqn (22) is now E = Eo - a - 2~ cos kpa = Eo a - 2 ~ cos * p. E = f (k p) and Z (E) = geE) are repre sented in Figures 1-20-a and 1-20-b. On ly one of the shared and adjace nt states in L4 = - and ~ = need s to be comp atibilised . It is also important to note that , in contrast to the middle band s (L2 and k2), the highest (L3, k3, ka) and the lowest (L I, k" ko) band s are well ' packed' due to the cosi ne form of the energy curve-flatte ned at top and bottom and near-vert ical aro und the ' waist' - from which they are derived . With increasing N, the phenomenon becomes ever more exaggerated, to the extent that Figure 1-20 would become overly com plex; with an increasing value of N, and therefore also kp, the result ing energy levels would be very tight in both the highest and lowest part of the band . Thi s qualitatively ex plains the for m of Z(E). In addition, the peak limit ca n be drawn for this functio n from the point at which the curve E = f (k ) reaches a tangenti al hori zont al. As Figure 1-20-b shows, the orbital density is highest at the extremes and least dense in the middl e of the band . Th ank fully, while with incre asing N the diagrams and functions become more and more co mplicated, the system shown here is a sufficiently simple exa mple !

-i

-i

3 Wavefunction forms for bonding and antibonding states With k = k p and N = 8, the wavefunction of eqn ( 10) ca n be written : \)J kp (x) Co L~=o exp (ikpta)\)Jo(x - ta ), with Co 1/ IN following normali sat ion . We can look at a representation of the function \)Jo(x - ta) = \)Jt (x) for s states (in which the wavefunctions
=

=

28

Optoelectronics of molecules and polymers

the state s characterised by 1 = O. Also, t ring closure, as detailed in Figure 1-19.

= 0 == 8 as these two values of t represent

a Atoms without interaction (for N = 8). By way of introduction, Figure 1-21 shows the general form of the wavefunction relative to the s states of a chain of 8 non-interacting atoms .

b Representation of the function (antibonding) bands

"'k

p

a-Low band states: p = 0 When p = 0, kp = 0 and exp(i kpta) the order of coefficient co): '1Jko = lflsl

(with N = 8) for low (bonding) and high

= 1, whatever value taken by t, we have (to

+ lfls2 + lfls3 + lfls4 + lflss + lfls6 + lfls7 + lfls8",O

A representation is shown in Figure 1-22.

Figure 1-21. Wavefunction of the s state for a chain of non-interacting atoms (N

x

I I

:lIll

= 8).

L= Nd

Figure 1-22. Representation of Wko'

Band and electronic structures in regular I-dimensional media

band states: p = 4. As 14 = i, successive values oft through kata = rrt and exptjnt) given in the table below:

29

~-High

t

kata = nt cos(rrt) Also '1..4 =

0=8 0= 4rr I

2 2rr

rr -I

I

3

4

3;r

4rr

-I

S Srr - I

I

= cos(rrt) are

6 6rr I

7 7rr -I

i: = 2a and the representation of Wk.t are presented in Figure 1-23.

c Bondingand anti-bonding states All atoms are in phase and all coefficients exp(i kota) are equal to I for the lowest band states, as shown in Figure 1-22. The resulting wavefunction 'lJkO exhibits no nodes as the probable electron density is the same between all atoms, explaining why this state gives rise to the strongest bonding. As energy increases, kp increases and the number of nodes in the wavefunction also increase. This can be seen in Figure 2 in AppendixA-2, in which the real part of the wavefunction Wk I corresponding to k] is shown. In the middle of the band, at which k2 = fa-, the states are neither bonding nor anti-bonding. This is shown in detail in Appendix A-2, Figure 3 for the real part of Wk2. However, for the highest level states, at the highest part of the band (14 = i and kaa = n ), successive values of exp(jnt) = cos(rrt) alternate between -I and I creating nodes midway between atoms (Figure 1-23). At these points the probability of electron presence is zero as they are distributed in an anti-bonding combination giving rise to an anti-bonding bond. As shown in Figure 1-24, the bonding states ('lJkO , 'lJkI , 'lJk_ d are lower in energy and more stable than the anti-bonding states ('lJk4, 'lJk3, 'lJk- 3). However, the energy levels corresponding to 'lJkp and 'lJk -p are identical (as kp = -k_ p, we have cos kpa = cos[k_pa], as Ekp = Ek-p ): these two functions are indeed associated in both senses of wave propagation due to the chain of atoms being a closed

4

-fll.~.l

(jls2

.....\

/"4 = 28

:::<jl!3.

•

(jl~~

.....•.

:::(jl.~5.....

(jls6

..::.<jl~.? . .. ...

•... .......

Figure 1-23. Representation of I\J~.

(jl, 8 es (jlso

30

Optoelectronics of molecules and polymers Eo+2~

'I'k4

Leveldegenerated 8 times corresponding to 8 functions CPst, witht= 1,2 ... 8=0

====

Eo- 2~cos(k3a)

:tb.-

Eo -2~cos(kla)

'1'''0

Eo-2~

Antibonding states (increasing energy)

1

l~i;~~f;'

10/"0- ) Figure 1-24.Levelsand states of energy for a chain of 8 atoms. ring . Just as for a free electron, we can make an appeal to physical solutions using cosines (Re{exp[ikpta]})-shown in Figures 2 and 3 of Appendix A-2-and using sine (Imlexpjikjtajj). These two types of solutions can be derived by using linear combinations (respectively, addition and subtraction) of Wkp and Wk-p. If, unlike in this example, the chain is not cyclic, then solutions of the form sin( J~1 can be obtained and used as coefficients for developing linear combinations of atomic orbitals (see, for example p. 41 of reference [Sut 93]). And indeed, we can see in this example that the interaction of neighbouring atoms results in an increased degeneration, giving 8 levels , each associated with a function of Wkp ' The degeneration is only partial as the two functions Wkp and Wk-p correspond to the same energy level. It is worth realising though that each function of Wkp gives rise to two additional functions, Wkp+ and Wkp. If each atom contributes only one electron to a bond, and for N atoms there are N electrons, then for our 8 atoms there are 8 electrons contributed which are spread throughout the available energy levels . And it is for this reason that only the 8 lowest states permitted are occupied, as shown in Figure 1-24. As the permitted band is only half-full, a conducting state is obtained if the span between the highest bonding level and the lowest anti-bonding level is small (i.e. N is large) . As we have seen , following Peierls transition, transitions from metallic to insulating states can occur.

)

4 Generalisation regarding atomic chain states In this Section VII, we have taken as example the s states, envisaging that they alone interact in a chain of atoms. In fact, the results we have obtained here can be extended to other states (most notably p states) . Further details on the more common systems are shown in Appendix A-2 .

VIII Conclusion The energy levels in a periodic network are determined by the extent to which the orbitals overlap, which is in turn controlled by the lattice constant a, on which the potential due to atoms is dependent, and the radius (R) of the outer atomic orbitals.

I Bandand electronic structures in regular l-dimensional media

31

When orbitals overlap sufficiently, such that they loose their individual identities (weak bonds with a < 2R), the semi-free model of electrons is particularly appropriate. To a zero order approximation, which assumes that the electron cloud is delocalised throughout a network , solutions for the wavefunction of form '" = A e± ikx are obtained, the ± appearing because of degeneration caused by the electronic wave being directed in one sense or the other. Alternatively, solutions of the form "'e = cos(kx) or = sin(kx), obtained from linear combination of the two exponential solutions, can be used. For an electronic wave with a large wavelength, or in other terms k = 2Jtj "A. is low, electrons are only weakly effected by the potential (V) generated by a chain of nuclei . In Figure 1-25 we can see that changes in V provoke negligible change in the evolution of the wavefunctions which have energy closel y approximating to that of a free electron (E = h2k 2 j 2m) and the probabilities of electron presence can, at the very limit ("A. -+ (0), be considered a constant throughout the network (i.e. density p = constant in Figure 1-5). However, when "A. is small , k comes close to the values expressed by k = ±Jt ja and the function s and tend towards functions ",+ and "'- (Section 11-2), which exhibit an evolution closely following changes in V and the probable electron presence becomes centred between or on atoms (Figure 1-5). The two functions are very different, as indicated by the resulting presence probabilities, and correspond to two different energy levels which are separated by a "band gap" of energy ~E = EG (see Figure 1-6). A more 'chemical' repre sentation can be given for strong bonds which are constructed from linear combination of atomic orbital s (LeAO), however, the space between the atoms remains important, in that each component orbital has an individual identity resulting from local variation s in the periodic potential. If combining orbitals are all of one type, the resulting energy develops in the form of a permitted band of height 4~ where ~ repre sents the coupling between adjacent atoms. If only s orbitals are considered, the lowest band states corre spond to bonds formed of bonding atomic orbital s, while the highest to anti-bonding orbitals (see Figure 1-26 in which there is one s electron per atom) . The resulting amplitude of the wavefunction is modulated by the exponential term exp(ikpta); when p = 0, the wavelength tends towards infinity (and therefore there are no nodal points within the bonding state) .

"'S

"'e

"'S

v x

IJIc

.: . . »-«:

---~----- ---IJIs --- ----_'. ~.

.

Figure 1-25. Wave function s \fie and \fis for large A..

x

32

Optoelectronics of molecules and polymers I isolated atom

2 atoms

4 atoms

N atoms antibonding orbitals

Es

!

bonding orbitals

Figure 1-26. Energy levels, without a band gap, formed by s electrons given by atoms in a chain. However, when k = rt/a, we have A = 2a and the atomic orbitals of two adjacent atoms are of opposite phase and the nodal points between them exhibit antibonding behaviour. This can be compared with Appendix A-2 (Parts I and II, respectively) in which the characteristic behaviour of a-s bonds and n-p bonds are discussed. For N atoms, N levels are realised, including N/2 bonding levels. As 2 electrons can be placed into each level (taking spin into account), the N electrons will fill the N/2 bonding levels, s electrons going into a orbitals and p electrons going into rt orbitals, leaving the N/2 anti-bonding levels empty. Can we also reiterate that for a given band, of size 4~, the higher the value of N, the more closely packed will be the resulting levels . We will see that the I-D model we have treated in this chapter can be extended to 3-dimensions in which the size and height of permitted bands are related to the co-ordination number, or rather number of bonds, of a given atom . We can also see that, as shown in Figure 1-16,if bonds between atoms in a I-D chain are alternating, then a central band gap is formed. Elsewhere, when different types of states are involved, a gap in the energy can occur due to the difference in energies of those orbitals, as shown in Figure 1-12. Moreover, when different orbitals-for example sand p type-mix, bonding and anti-bonding orbitals are formed, and the difference in energy bands corresponds to the energy of the formed band gap. We will look at this problem in 3-D, including the formation of hybrid orbitals-for example sp3-in Chapter 2.

II

Electron and band structure in regular or disordered 3-dimensional environments: localised and delocalised states

I Introduction Calculations based on 3-D environments, using weak bonding approximations, follow much the same line as the studies made in I-D. The dispersion curve E = f(k) can be traced depending on the different directions under consideration (kx , ky and kz for a cubic crystal). If these directions are not equivalent, and have a forbidden energy for which the value is direction dependent, then the resulting energy gap in the material is of the form. EG = (Ec)min - (EY)max in which (Ec)m in corresponds to the minimum conduction band (CB) for all directions k considered, and (EY)max corresponds to the maximum valence band (VB) over all direction s. This approximation for the weak bond is, in fact, only applicable to metals. In this Chapter, we shall look at the electronic bands found within 3-D organic solids and see that their intrinsic semiconducting or insulating character can only be realised by considering strong bonding. We shall also consider 3-D regularly networked solids, considering each node an atom which contributes to the electronic propert ies of the material, via: (i) a single, s-state electron , and using as example the cubic network to determine the height of the permitted band , otherwi se known as the VB. (ii) hybridised electrons using the specific example of diamond , in which each carbon atom is at the centre of a tetrahedron and has sp3 hybridised bonding states (as detailed in Appendix A-I , Section II-2); the generation of the band structure and the forbidden band, which separates bands corresponding to bonding and anti-bonding states, will be described . Finally, we will look at amorphous material s.

34

Optoelectronics of molecules and polymers

II Goingfrom I-D to 3-D: band structure of networked atoms with single, participating s-orbitals (including simple cubic and face centred systems) 1 3-D General expression of permitted energy To simplify eqn (22) in Chapter I, which relates the energy of a strongly bonded electron in I-D, we can rewrite it as

L

E = Eo - a - ~

e -ikla

(1)

1=-1,+1

The sum is for one atom and its two closest neighbours. On considering more than one dimension though, a simple way of writing eqn (1) is E = Eo - a - ~

L e-ika;;' ,

(2)

m

in which a~ represents the vectors joining the reference atom with its m closest neighbours. In the case of a cubic lattice, as shown in Figure II-I , the closest neighbouring atoms have vector a~ components: (±a, 0, 0)

in the x axis

(0, ±a, 0)

in the y axis

(0,0, ±a)

in the z axis

1

The energy thus takes the form E = Eo - a - 2~[cos kxa + cos kya + cos kza], in which kx, ky, kz are the components of k in the 3 directions Ox, Oy, Oz. In the centre of the zone, k = ko = o(or k, = ky = k z = 0); the energy is minimal and equal to: (3) E = Eo - a - 6~ = E(ko) In the neighbourhood of the central zone, k cos kxa ~ 1 -

(k xa)2

2

~

ko

~

0, or k x ~ k y ~ kz ~ 0,

(ditto for k y and kz), and the energy can thus be written as E = Eo - a - 6~ = E(ko)

+ ~a 2(k; + k; + k;)

+ ~k2a2.

Figure II-I. Geometry of cubic lattice structure.

(4)

II Electron and band structure

35

We can now compare eqn (4) with that obtained by a Mac Laurin development of E using ko: 2E) (k - kO)2 (a 2 ak2 ko '

aE)

E(k) = E(ko) + (k - ko) ( ak ko +

(5)

As in the centre of the zone , we now have a tangential horizontal (to be compared with Figure 1-10), (~;)ko = 0 and the mas s can be effectively defined by

n2

(6)

m"= (o2E ) ' ok2

= 0:

so that we can obtain from eqn (5), when k :::::: ko

,,2

E(k) = E(ko) + (k - kO)2 2m *

(7)

Again, in the neighbourhood of the central zone, where ko :::::: 0, we now have E(k) = E(ko)

n2

+ _k2

(7')

2m *

Comment: The mass m of an electron can be related in the fundamental dynamic equation FT = q(Eappl + Eint) = my with Eappl which is the empirically applied field and y is the acceleration undertaken by the electron. The internal field (Eind which is derived from the internal potential generated by the nucleus is not well known and the effective mass m" is defined by the relationship Fext = qEappl = m*y. Eqn (6) is obtained by calculating the work of the external force (see Appendix A-2, Section IV-2).

2 Expressions for effective mass, band size and mobility The identification of coefficients k" (n

= 0 and n = 2) of eqn s (4) and (7) yields :

E(kO) = Eo -.

(1-

~a2 = - - ,

or

j

n2

2m *

6~

(8)

m"

The size of the band can be deduced from the amplitude of the variation in energy in the first Brillouin zone (described, for I-D, as the variation in k zone in Figure 1-10): -

when k, when k,

= ky = kz = 0 : E = E(ko) = Eo = ky = kz = .; - : E = E(';;-) = Eo -

(1 (1

6~ ;

and

+ 6~.

The amplitude in the variation (4~ in I-D) ofE, which is the depth of the permitted band, changes in 3-D to

(9) The result given for a simple cubic network can be generalised by introducing a co-ordination number Z, which denotes the number of closest neighbours, and in

36

Optoelectronics of molecules and polymers

our example it is equal to 6 as made evident in Figure II-I. Eqn (9) can thus be rewritten as (10) B = 2Z~. Interestingly enough, when mobility is expressed in the form duction of'B derived from eqn (10) of jJ." we have

jJ.,

(~ = B/2Z) in eqn (8) gives: m"

= ~~, the intro-

= ~Z. In terms (11)

We can thus conclude that semiconductors have narrow permitted bands as .6.E = B is small. There is weak coupling between atoms as, following eqn (I 0), ~ is also small; in other words , semiconductors display low mobilities.

III 3-D covalent crystal from a molecular model: sp3 hybrid states at nodal atoms 1 General notes We will now look at the case of diamond, a material made up of a regular network of sp3 hybridised carbon atoms. In Appendix A-I, the spatial geometry of bonded carbon is detailed. The bonds are equally spaced when sp3 hybridisation occurs and the orbitals can be expressed using 4 functions 1\111 >, 1\112 > , 1\113 >, 1\114 > as calculated in Appendix A-I , Section III. To follow the formation of the different electronic states and energy levels in diamond, we will sequentially study each step as shown in Figure 11-2, by:

a Isolating carbon atoms Looking at Figure II-2-a and Figure 11-2-b, zone (1), we see that the orbitals of isolated carbon atoms , with electronic configuration ls2 2s 2 2p2, are characterised by having two levels, E, and Ep . Note that in Figure 11-3, the atoms C, C', C" and so on, are assumed to be well separated. As we bring the C', C", CIII , C"" closer to the reference atom C, sand p bands form following the superposition of wavefunctions. For example, s-orbitals give rise to bonding and antibonding combinations which tend downwards and upwards, respectively. This is detailed further in Appendix A-2, Section 1-2.

b sAnd p band hybridisation at the critical point M of Figure 11-2(a) When hybridisation of sand p states is energetically favoured, sp3 states on atom C, described by functions \111, \112, \113 and \114, are obtained (see Appendix A-I, Section 11-2). In the same way, hybrid states of the atom C' are represented by the functions \II;, \II;, \11 and \II~, and so on, for the other C atoms .

3

II Electron and band structure

37

(a)

(a)

Interaction energy \

\ I \ \ I \ \ \ I \ \ I \ (2) E*chatom C, \ \ C'... ~ hybr;ised sp3

(b)

s, = Esp3 EG)final=2~-4~

Electroniclevels sand p from isolatedC atoms (configuration 2s' 2p')

Hybridised states of C'I'I, 'l'z, '1'3' '1'4 Same level, Eh, for hybrid states '1" I ' 'I" z,'1" 3, '1"4 ofC'. Ditto for Cn, cr. C''''. (See Figure 11-3 for definition of C, C, C" ,C"' ,C'"')

4~

Bonding states $1Land antibonding states 'PIA hetween e and C', $2L and $2Abetween e and $3Land $3A between e and em. $4Land $4Abetween C and C''": in lotal, 4 bonding orbitals $Land 4 antibonding $ A

cr.

associated with bondin g

levels EL and antibonding Ek

Increasing degeneration due to intervention from couplingbetween neighbouring bonding orbitals(
Figure 11-2. (a) Band format ion due to clos ing C atoms ; (b) evolution in electronic energy levels through successive couplings.

Taking all states together, as shown in zone 2 of Figure II-2(b), and represented by the functions \IIi)i=1 ,2,3,4, \IIDi=1 ,2,3,4 and so on , equivalent to 4N states for a system containing N atoms, we have energy E sp3 = Eh.. Eh can be calculated relatively simply using, for example Eh = ( \II JlH I\II I )

= G(S

+
I

= 4{(SIHIS) + (
I

4{Es + 3E p },

38

Optoelectronics of molecules and polymers

e"" em

\ttII~-----y

Figure 11-3. Relative locations of atoms and available sp3 couplings.

with

Ep and Es , respectively, represent the energy levels of 2p and 2s states shown in Figure 11-2-b. Thus

(12 )

c Type A couplings between neighbouring C atoms, as detailed in Figure 11-4 Here we con sider only couplin gs (I), (2), (3) and (4), otherwise noted as C - C' , C - C" , C - C"', C - C"" , assuming that interactions that could result from other bonds are negligible . The bonding and anti-bonding states appear as shown in zone 3 in Figure II-2-b. This is detail ed, qualitatively, in Section 2 ju st below.

d Supplementary effects resulting from type B couplings between molecular orbitals, as shown in Figure 11-4 Type B couplings are those between (I) and (2), between (2) and (3) and so on, and result in the appearance of energy bands as shown in zone 4 of Figure II-2-b. A quantitative approach is detail ed in Section 3 below.

2 Independent bonds: formation of molecular orbitals Zone (3) in Figure 11-2 (b) shows the states which appear following coupling of two sp3 hybridised orbit als, for example, of C and C'. For this atomic couplin g, solutions can be given in the form of a linear combination of each atom's orbitals. As orbital 1\f1} is for atom C, and orbital pl") is for C' , we now have = c l\f1} + c'I \f1'}.

II Electron and band structure

39

c" 'l' 2"

I (2) 'l'

'l'

rrr

:

~

Coupling between bonds

1'l'2

C'" _ 3 _(3) _ _ 3 C

- - ( 1 ) - - X'

I 'l'1<;

'l' 4

(4)

A -,,'l'I' 'l' = <'l' JlHI'l'I'>

I'l'/'"

e"

ll

Figure 11-4. Representation of successive A and B couplings by projecting plan view of FigureIII-3. On using line (1) in Figure II-4 to indicate the bonding between 2 C atoms, the resulting molecular orbital (
(13)

By taking into account pairs belonging to each carbon atom, and assuming them to be independent, the molecular orbitals which appear about our reference atom C are, in addition to
=

I

/I

y'2(1 1IJ2} + 11IJ 2})

I /I I

and

»

and

~(I11J2) + IIIJ~/I})

and

The energy levels EL and EA are , respectively, associated with bonding and antibonding states . They have the same form as that determined in Appendix A-I, that is: EL = E~ - ~ and

EA = E~ +~ .

(14)

Note that E~ = Hjj = (lIJil H llIJ j) = Eh = Esp3 and that the coupling parameter between two atoms under consideration is -~ = Hjjl = (lIJil H IIIJ; ).

Comment For N atoms in a crystal, the number of bonds of type
40

Optoelectronics of molecules and polymers

between 2 atoms . However, the actual number of valence electrons per atom is 4, resulting from 2s z2 pz -+ 2t 4 , in which t represents hybrid states, and therefore the fundamental state carries 4N electrons, which can also be written as 2 x 2N (the number of electrons per bond multiplied by the number of bonds). All bonding bonds are therefore full when the symmetrically numbered anti-bonding bonds, A, are empty.

3 Coupling of molecular orbitals and band formation For a crystal containing N atoms, following the reasoning of Section 2, the energy level EL (and EA) is degenerate 2N times, corresponding to 2N bonding orbitals. We will now look at the effect of coupling between different molecular bond s on the degeneration of energy levels.

a Effect of coupling energy between hybrid orbitals on the same carbon atom The coupling energy between two hybrid atoms is of the form: (\lII IH I\lIz) terms of either \lII and \liz (given in Appendix A-I , Section 11-2-c) : -/1 =

(~(S + X + Y + ZIHI~(S I

= ::leEs - Ep

-

X- Y

= -/1. In

+ Z»)

I Ep + Ep ) = ::leEs - Ep ) .

We can see that the effect is not zero and that we should therefore expect, for a covalently bonded 3-D crystal, a non-zero coupling effect between molecular orbitals bonding two adjacent atoms .

b Coupling effects between neighbouring bonding orbitals within a crystal matrix The coupling within the crystal matrix , shown as B in Figure 11-4, corresponds to the form

Ignoring coupling integrals between non-adjacent electrons, as for example (\lI; IHI\liz) ~ 0, results in:

1

I

I

( ILIH Izd = 2" (\lIIIHI\lIz) = 2" x ::leEs - Ep) =

/1

-"2'

(15)

On modifying energy levels of type EL, coupling of molecular bonds results in an incre ased degeneration to the level of EL = Eb - ~ . By analogy to the I-D system looked at in Chapter I, wavefunctions of the crystal must be written in the form of a linear combination of either bonding orbitals, L, or

II Electron and band structure

41

of anti-bonding orbit als, A. These functions, which are charac teristic of a regular network , should also satisfy Bloch 's theorem and are thus of the form:

I tcr)}= Co L eik.rs I LCr - rs)) s

1 ~(r )} =

c'o

L eik.rs I A(r - rs)) s

These are the Bloch sums for bonding and anti-bonding orbitals, and are for electrons delocalised throu ghout a whole crystal network in 3-D. Th ey are similar to the wavefunctions used in 1-0 to verify Floquet's theory (eqn ( 10) of Chapter I). In the same way as that observed for s-orbitals in a \-0 system (Chapter 1) and in a 3-D system (Section II-I of this Chapter), the functi ons result in a fragmentation of EL and EA levels, as show n in going from zone 3 to zone 4 in Figure 11-2. We thu s obtain 2N Ibr)) func tions. Having taken into account 4N spin functions we now have a full band of bonding states, j ustifying the term "Hig hes t Occup ied Mole cular Orbital" (HOMO), otherwise known as the valence band by physicists. The band of anti-bonding states though is empty and is kno wn as the "Lowest Unoccupi ed Molecular Orbital", or for physicists, the conducting band. The pair of band s are separa ted by what is known as the "band gap" of height EG. Quantitatively, we have see n in Sec tion 11 of this Chapter that for s-orbitals char(WsIH IWs±l)) , the size of the formed band s acterised by a coupling parameter - ~ is equal to 2Z ~ (eqn (10) of Section 11). In the case of Figure 11-4 treated here , the co-ordinat ion number (Z) equals 4 and the coupling parameter, ( lLIH I2d = - ~ . The size of the HO MO and LUMO bands is therefore, following eqn (10), B = 2.4 . ~ = 4Li (zo ne (4) of Figure 11-2-b). In additio n, the height of the band gap can be calculated dire ctly fro m Figure 11-2-b using EG = 2~ - 4Li . Th e values of ~ and Li depend on the network and size of atoms. With diamond having a band gap of aro und 5.4 eV, it is more of an insulator than a semico nductor. On descending down through column IV of the peri odic table, moving fro m carbon, throu gh silicon to germanium, the size of the atoms increases and the size of the perm itted band s also increases to the order of ca. 4Li. With eac h succe ssive increase in atom size, the band gap diminishes: C, 5.4 eV; Si, 1.1 eV; and Ge, 0.7 eY.

=

IV Band theory limits and the origin of levels and bands from localised states 1 Influence of defaults on evolution of band structure and the introduction of 'localised levels' Figure 11-5 continues on from Figure 11-2-b by considering the origi n of the VB and CB for a perfectly ordered system of tetrahed ral carbon atom s. As we have see n, the initial s2p2 co nfiguration gives rise to 4 sp3 type molecular orbital s. And eac h one of

42

Optoelectronics of molecules and polymers E

....... 2p' H

ltlt

Dangling bond

2s' H

a

Figure nos. Origin of localised levelsassociated with dangling bondsof tetrahedral carbon.

these leads to the formation of a bonding a-orbital and an anti-bonding o" -orbital , In going from single molecules to the solid state, the combination of sp3 orbitals leads to the rupture of a- and anti-bonding o" -orbitals into valence and conduction bands, respectively. Because of the finite size of a real crystal, however, at the surfaces faults occur as each carbon atom is bonded to 3 rather than 4 carbon atoms. This results in one incomplete sp3 bond, or "dangling" bond, which contains one electron and, intrinsically, is electrically neutral. The single electron is situated at the level E sp3 , even if the localised level associated with the electron is ELoc, and is in the middle of the band gap, given the permitted bands allowed (Figure 11-5). Other faults can give rise to similar levels in a real crystal: vacated sites (generated during the preparation of the crystal); and dangling bonds induced by physical treatment, such as irradiation or ion implantation which , breaks bonds as the crystal is traversed. The presence of structural faults, caused by dangling bonds, can create disorder, for example fluctuations in bonding angles, and result in an opening of levels and the formation of a default band. The exact positioning of the bands relies on relaxation phenomena which occur in the solid following fault formation, and whether they result from valence or conduction bands. In Figure 11-5, the lower band, near the middle of the band gap, corresponds to a dangling bond containing one electron. It is therefore a donor type band which is neutral and in an occupied state. The upper band , near the middle of the band gap, corresponds to the same fault but has a different charge i.e. has received an extra electron, and is an acceptor band which would be neutral if it were empty (see page 344 of [Ell 90]). The energy difference between these two types of faults, of which one is neutral when it is full, the other neutral when it is empty, corresponds to the Hubbard correlation energy (U), for which U = (q2j4moErrn) , in which rl2 designates the average distance between two electrons on the same site over all possible configurations.We

II Electron and band structure

43

will now go on to try and detail the effects resulting from these electronic repulsions which up until now have been treated as negligible .

2 The effects of electronic repulsions, Hubbard's bands and the insulator-metal transition In band theory, until now, we have considered that each electron existed in an average potential resulting from a collection of atoms and other electrons. In the case of alkali metals (Li, Na, K. .. ), which have one free electron per atom, the transfer of an electron from one atom to its neighbour through a conduction band occurs via electronic levels situated just above the Fermi level (EF) and the energy utilised is extremely small , of the order of a fraction of a meY.

a The model In utilising Hubbard's model and theories, we can consider that the only important electronic repulsions are those which occur between two electrons which are on the same site (the same atom in a series of alkali metal atoms) . The repulsion energy, or Hubbard energy, can be evaluated to ascertain if it is significant for certain materials and can even help indicate the origin of certain metal-insulator transitions. As before , we will use the same chain of alkali metals as shown in Figure II-6-a to evaluate the problem, although we will assume that overlapping between atoms is poor and the transport of electrons from one atom to the next requires a great deal of energy. Movement of an electron thus generates a supplementary repulsive energy which can be estimated by: -

calculating the ionisation energy required (lp) to separate an electron from the atom A~ to which it is attached which subsequently becomes Al (this change is shown in going from Figure II-6-a to Figure II-6-b); and

W

t + t

N

+ t

~

+ t

+ t

888888888 Figure 11-6. Highlighting electronic repulsions in a chain of atoms with s-orbitals.

44 -

Optoelectronics of molecules and polymers calculating the energy recovered, or the electron affinity (X) when the free electron is placed on the independent, adjacent atom A;, which subsequently becomes A2.

The total energy thus required, equivalent to the repulsive energy, is UH = Ip - X. For hydrogen Ip = 13.6 eV and X = 0.8 eV, and thus UH = 12.8 eV, showing how UH can attain a relatively high value of several eV. Elsewhere, Mott showed how the repulsive energy can be calculated using fJ2 [Mot 79], which represents the distance between two electrons on the same site or atom, and "'(r) which is the wavefunction corresponding to the value proposed at the end of the preceding Section I .

UH =

ff

2 e ,,,,(rd,2',,,(r2),2 dr, dr2. 41t€orl2

Practically speaking, this energy is particularly important with respect to transition metal oxides, such as NiO, for which electron transport occurs via d-orbitals and can be written as: NiH + NiH ---+ NiH + Ni+ For a chain of alkali metals, however, the same electron transfer, via s-orbitals, is written : Cs + Cs ---+ Cs+ + Cs" In Figure 11-6, A; == A; == Cs while Al == Cs" and A2 == Cs". On removing the arrows in Figure 11-6, which represent the division of electrons throughout a chain of atoms , we can consider that for NiO, A; == A~ == NiH , A] == NiH and A2 == Ni+. Placing an electron on a NiH, to form a Ni+ ion, would require the energy given by UH = I - X if the Ni+ and NiH ions, at positions A2 and Al in Figure 11-6-b, respectively, are sufficiently far apart. The transported electron can be assumed to pass through a free state, that is its energy En at the level n --+ 00 tends towards 0, as do the successive energies Ip and -X, as previously described. Energy levels of isolated ions can be represented in terms of -Ip (the energy of an orbital which looses an electron , i.e. NiH or Ad and -X (the energy of a supplementary electron situated on Ni+ or A2)' When the ions are well separated , as shown in the far left part of Figure 11-7, each energy level is separated by U H = Ip - X which appears as a band gap between the upper and lower levels, the former having received an electron, the latter having lost one. On bringing the ions closer to one another, as described in going from the left to right side of Figure 11-7, transport by charge carriers becomes possible via the permitted bands which start to form. These newly formed discrete bands give rise to permitted bands (Hubbard's bands), upper level bands of electrons (in which Ni+ can be found) and lower level bands containing holes (in which NiH resides). As the size (B) of the bands grows with increasing proximity of atoms, the difference UH - B decreases and eventually disappears when Breaches UH. Beyond this value--obtained when the atoms are close enough to each other-the upper and lower Hubbard bands overlap and the band gap is removed ; this point is also known as the Mott-Hubbard transition from an insulator to metallic state.

II Electron and band structure

45

E

o

-x [p

N1·3+. - IP

Insulator

B=U

H

Metal

B

Figure 11-7. Evolution of Hubbard bands as a function of band size (B). B = 0 for atoms far apart but when B = UH, the band gap UH - B disappears to give a metal-insulatortransition.

b Charge transfer complexes Charge transfer complexes (CTCs) are materials in which the effective correlation energy is high [And 92]. If the effective energy (Ueff) is defined as the difference between the electronic repulsion energy for a site occupied by two electrons (U o ) and the electronic repulsion energy between two electrons on adjacent sites (U I) i.e. Ueff = Uo - UI, then for a CTC the energy UH corresponds to Ueff. For a system with N sites: -

-

-

if we can assume that Ueff is negligible, each site can be occupied by two electrons (spin up, t, and spin down, .J,.). In addition, as in Figure 11-8-a, if the system is half filled by N electrons then the material is metallic; if the system is one in which Ueff is high, we can place only one electron per site. Again, if the system carries N electrons (i.e. p = I, in which p designates the number of electrons per site) then all energy levels are occupied and the band is full as shown in Figure 1I-8-b. Only B inter-band transitions are allowed, demanding a high energy of activation (Ea), and the system, in other words, is an insulator (Mott insulator) or semiconductor. For example, the complex HMTTFTCNQF4, in regular columns, has p = 1, E, = 0.21 eV with a room temperature conductivity aRT = 1O-4Q - 1 cm- I ; and once again, if the system is one in which Ueff is high and we can only place one electron per site but p < 1 because bonds at the interior of each column are not fully occupied, both A intra- and B inter-band transitions are possible with the former requiring, respectively, low and high activation energies. This is shown in Figure 1I-8-c. As an example, TTF+ o.59 - TCNQ-0.59 displays a metallic character with p = 0.59 and aRT = 103 Q - I cm" :

46

Optoelectronics of molecules and polymers E

E

t1111

E

k

k

A

(a) Ucf('" 0 : electrons in 2 states, Tand T. allowe d per site . If sys tem has N sites and N electrons. ther e is a semi- full band and a meta llic state.

(b) UcfThigh. and on ly 1 electronic sta le allowed per ire. If sys tem has ' sites for ' elect ron s (p = I). there is a full ban d and the material is an insulator or semiconduc tor and only inter-ban d B transitions are allowed.

(e) efT high. and on ly I electron ic state allowed per site, If P < I. band partia lly full and bot h A intra- and B inte r-ba nd transitions are possible.

Figure 11-8. Electrontransportwith respect to electronic structure. Upper parts of the Figures representband schemesand lower parts representelectron positions (e = occupied state, 0 = empty state).

c The Mott transition from insulator to metal : estimation of critical factors Different theories have been elaborated to establish, in a quantitative manner, the parameters surrounding transitions from insulator to metall ic states . The ThomasFermi screened potential can be used [Ell 98], [Sut 93] and the basis of theoretical developments, including the application of magnetism, can be followed up elsewhere [Zup 91]. We will limit ourselves here to saying that this transition can result from competition between localisation effects, themselves resulting from electron Fermi kinetic energies (EF) and the electrostatic energies to which they are subject. In order to take into account environmental effects and polarisation, two elements must be considered: the permittivity of the medium under study (EoEr)(Er being the relative permittivity of the material); and the active length (a*) of the electrostatic potential, which takes on the form e 2 / 4n EOErKl a" . KI is a constant which accounts for the present-day incomplete knowledge of interaction distances, which can be written simply as Kj a" . It should be noted that a" must take on the same form as the first Bohr orbit (ao), that is ao = [Eoh 2] /[nme 2 ] . To obtain a" from ao, EO needs once again to be replaced by EOE r , to take into account the effect of the interaction between network and electron thus changing the latter mass from m to m" so that a" = [ErEoh 2] /[nm*e2 ] or a* = Erao(m/m*).

II Electron and band structure

47

As EF = (h 2/2m*)(3n 2n)2/3 = (K2h2n2/3) /(4n 2m*), in which n is the electron concentration (page 171 of [Mooser 93]) and K2 a constant, the condition required to reach the metallic state can be written: (h 2n2/ 3)n=ncl(4n 2m*) ?: Ce2/(4moEra*), in which C is, as yet, an unresolved constant resulting from the introduction of the aforementioned constants K, and K2. The relationship is thus based on nc, at which point the transition occurs. Therefore , we should have a*n; /3 ?: C(e 2m*n) /(EoE rh 2), either in terms of the expression s a" and ao, as in (a*n~ j3)2 ?: C, or expressed as a*n~ /3 ?: D, in which D = C' /2. Experimentally, the constant D is normally found to be around 8 times the value of nc (it has been shown that D ~ 0.26) and thus the criteria for the transition is: (a*n~ j3)2 ?: 0.26. (16) Physically speaking , this criterion means that all materials can become metallic if they are sufficiently compre ssed so that the electron density reache s the value ncoThe corresponding metal-insulator transition (M-I transition, which also occurs at n = nc) is called the Mott transition and originate s from localisation of electrons through electro static interaction s, not from any material disorder. We shall see in the following Section 3 how disorder alone can result in the Anderson transition .

d n-Conjugated polymers Polymers conjugated by n-orbitals are, in principle, not subject to Mott transitions as transfers from one site to another in the same chain have ~ integral values which are too high (typically of the order of 4~ ~ 10eV for polyacetylene), well above electron-electron interaction energies (U, below 1eV for polyacetylene). Figure Il-8-a therefore sufficiently describe s these material s, although they do display insulating characteristics, which in the case of polyacetylene results from a Peierls distortion due to electron-phonon interactions which open the band gap (Figure II-9).

3 Effect of geometrical disorder and Anderson localisation a Introduction The effect of geometrical disorder has for the most part been studied within theories on amorphous semiconductors developed by Mott and Davies [Mot 71, Mot 79 and Mot 93], and discussed-in French-by Zuppiroli [Zup 91] and Moliton [Mol 91]. The theory is based on two fundamental ideas: -

the first was taken from the work of loffe and Regel [Iof 60] who observed that there was no great discontinuity in the electronic properties of semi-metallic or vitreous materials when going from solid to liquid states. It was concluded that electronic properties of a materials cannot be only due to long range order, as was proposed by Bloch for properties of crystals , but are also determined by atomic and short range propertie s in which the average free path of an electron is interatomic . It is worth noting also, that even though a material may be amorphous,

48

Optoelectronics of molecules and polymers

~~5-::I:~~~sc Molecular state

Metallic solid state; right zone of Figure II-7

Band gap opens following network distortion: lower band is full

Peierls transition _

~

Insulator state

U » 4 ~: scenario from Figure II -8-a

(b)

-L::~ ~ ------- ------rp Molecular state

Solid state; left zone of ~ Figure II-7

Band gap opening following strong electron-electron interactions: Molt transition

Insulator state

Figure 11-9. (a) Characteristics of n-conjugated polymers, with the exampleof polyacetylene undera Peierlstransition, verifying that U » 4~ (and possiblefor CTCs).

« 4~, in contrastwith (b) Mott insulators for which

U

-

this does not exclude it from having bands. For example, glass, which is a noncrystalline material, is transparent in the visible region of light (:::::: 1.5 - 3 eV), that is to say that while absorption of photons with energy below 3 eV does not occur, glass does actually have a band gap of at least greater than 3 eV; and the second rests on the evidence given by Anderson [And 58] for a material without long range order that nevertheless have localised states with permitted energy bands for electrons. This theoretical model comes from observations made on certain amorphous semiconductors in which charge carriers cannot move .

b Limits to the applicability of band theory and Ioffe Regel conditions Bloch functions, i.e. Wk (r), can be used to describe electron wavefunctions in perfectly crystalline materials. The electronic states are delocalised and spread out over space, as denoted by IWk(r)1 2 . Because of perfect delocalisation, the average free mean path of an electron can be considered infinite. It is only when studying a real crystal that the average free path of an electron takes on significance because of effects due to quasi-imperfections caused by vibrations, called phonons, and imperfections caused for example by doping agents and impurities which perturb the regularity of potential throughout the network. It is only when these electron scattering effects, which limit the free path of electrons are considered, that the statistical average term L of the free path length of an electron between two successive collisions can be introduced. In addition, there are two term s to note : "lattice scattering" which indicates collisions due to the material network and for a similar effect caused by ionised impurities, the term "impurity scattering" is used . On disordering a lattice by introducing vibrations and /or impurities, L appears and takes on a decreasing value as disorder increases. If there is a low amount of

II Electron and band structure

49

impurities, then local levels appear, most notably in the forbidden band (FB), but if the number of impurities increases, the localised levels grow to form impurity bands which can reach a size D.Ee , close to that of the valence band (VB), the conduction band (CB) and the FB introduced in Bloch's theory. Bloch 's theory though looses all semblance of reality when values of D.Ee reach the same values of the bands . Put another way, we can go from the crystalline state to the amorphous state with L decreasing until Bloch 's theory is no longer acceptable. The limit for L was fixed to k£ '" 1 (for a perfect crystal, k£ » I) by loffe and Regel by following the reasoning of the uncertainty principle, i.e. D.E . D.t ::: Ii

and

D.x · D.k ::: 1.

(17)

To arrive at the result shown above, we can consider that the trajectory of an electron after a collision is random, and at the very best can only be defined between two collisions, i.e.: (18)

(D.t)max = t in which

t

is the relaxation time-the average time between two collisions; and (19)

From eqn (17) we can thus directly derive the best precision in D.E, (D.E)min , and in D.k, (D.k)min, when : 1) the equivalence of (17) by D.E is verified. D.t = Ii and D.x . D.k = 1; 2) when D.t and D.x are at their highest value in the equalities just above and equal to (D.t)max = t and at (D.x)max = L . We arrive at: (D.E)min . t

~

It,

(20)

and (21)

The question we are therefore brought to ask is with increasing disorder, what are the lowest values that t (and thus the mobility I.t = qr / m) and I: can go to while (D.E)min and (D.k)min retain acceptable values, values which are compatible with classical theory of bands in a real crystal. The response can be given be using simple calculations which show that when : •

I.t ---+ I cm 2 V-I s-I (and t ~ 6 x 10- 16 s), from eqn (20)( D.E)min ~ leV. Thus,

(D.E)min ~ EG (band gap size) or (D.E)min is the same order of size as the permitted bands . When I.t S 1 cm 2 V-I s-I , incertitude in the energy of the carriers tends to the same order of size as the permitted and forbidden bands , i.e. to such an extent such that the band scheme looses its relevance to real systems. It will be shown though that the Anderson model band scheme has to take into account localised bands with a gap which eventually becomes the mobility gap, Ell'

50

•

Optoelectronics of molecules and polymers

£

~

a few Angstroms, that is to say £ ~ a, in which a is the lattice constant and is typicallyoftheorderof3 x 10-8 em, eqn (2l) results in (llk)min ~ 1/£ ~ l la ~ 3 x 107 cm- 1 ~ k .Ineffect, with A. = h/rnv in which v = Ythermal ~ IOOkms- 1 and A. ~ 7 X 10-7 cm, we have k = 2rr/A. ~ 107 cm"", and can directly infer that in a band scheme , conduction electrons will be such that k ~ l /a. At these values where £ ~ a, we therefore have (llk)min ~ k, and k can no longer be considered a good physical parameter to which we can apply quantification. In addition , when Ak '"" k Fermi's sphere is so badly defined that it can, at a limit, be totally deformed and the concept of carrier speed looses significance as hk = m'v, ju st as much as the average free path which is expressed as a function of v following

£

=Vt.

Finally, as soon as L ~ a, and more strictly speaking as soon as E :s: a which occurs when the interaction between an electron and the material network becomes increasingly strong, an electron no longer goes any further than the limits of the atom to which is tied. The electronic wavefunction localises over a small region in space and is generally supposed to diminish exponentially with respect to R following exp(-aR) . Having followed the work of Mott and Anderson, we are brought to a new concept of localised states. The permitted density of states, N(E), always result s in an energy band beneath a single Ec for a conduction band and above a single Ev for a valence band , and, in other words, an activation energy is necessary for carriers to pass from one state to another with an emission or absorption of a phonon .

c Anderson localisation a) The model Systems in which disorder is due to a random variation in the energetic depth of regularly spaced sites (with interstitial distances always equal to a) are considered in Anderson's model, and can relate, for example, to a random distribution of impurities. Different authors, including Mott, have tried to take into account lateral, spatial disorder and the results have been close to those of the Anderson model , of which we will limit our discussion to this section . In Chapter I, we saw that if we take into account effects resulting from a network of atoms at nodes by constructing a regular distribution of identical potential wells, then a permitted energy band of height B appear s, as shown in Figure 11-10.

a

E B -----------~~~ - ----- -----~~--'

N(E)

Figure 11·10. Regular distribution of identical potential wells and permitted band.

II Electron and band structure

51

In the approximation of strong bond s, we saw in Section II - I of this Chapte r that: •

B = 2Z ~, with Z = the number of adjacent neighbours and integra l between two adjacent sites;

~

the resonan ce

• (22) 2

so that I-l = q~~ ~ , and semiconductors possessing a narrow B band exhibit low mobilities. For Anderson's model we repl ace the precedin g distribut ion by one of randomly deep potential wells which repre sent disorder, as shown in Figure II-II . ~)

Variation in wavefunctions with respect to VoiB (Anderson) and [, (Joffe and Regel)

We will show here how perm itted energy bands chan ge into localised states if Vo/B goes beyond its critical value. In order to do this, we need to look at the following, success ive scenarios:

-

Real crystal: Vo/B is very low and [, is high

Here the wavefunction is given by Floquet' s theory (eqn (10) of Chapter I) which we can write to the order of a normalisation constant: l\Jk(r ) =

L eikrnl\Jo(r -rn ).

(23)

n

The average free path can be estimated from the Born approximatio n, [page 40 I of Smi 61], by realising that the wave vector of an electron (k n) changes to k m once the electron has undergone a co llision of probabilit y Pmn and that P nm = ~ = and in addition [page 16 of Mot 71] P nm is given by Fermi's 'golden rule' : P nm = I Q I~oy N(E m) (eqn (2.20) of [Mol 9 1J relating to unit volume). For conducting electrons, for which Em ~ Er, spread throughout a volume V = a 3 with a random

t,

Ph

v E

v, {

Figure 11-11. Distribution of randomly deep potential wells.

52

Optoelectronics of molecules and polymers

distribution of wells with depth such that IQ Imoy = (Vo/2), we obtain:

..!.. J:

= Pnm = ~ 2n v

41i

(vo)Z a3N(EF) 2 v'

(24)

in which N (EF) is the density of states at the Fermi level and v the velocity of an electron at the Fermi level. As J: is large, the system under consideration is almost a perfect crystal and therefore we can write: _ (2E) l iZ _ 4n(2m*)3 I ZE I / Z N(E) 3 and v h m* In using the effective mass given in eqn (22) , eqn (24) gives: a

(Vo/~)Z . 32n

-

'

(2ZVo/B)z a B = 2ZI, - = -'------'----J: 32n

. and with

(25)

System in which (VolB) ~1 (single disorder value) corresponding to J: ~ afor weak disorder When J: ~ a, eqn (25) written for a cubic system in which Z = 6 results in (Vo /B) = 0.83 ~ 1. At this point when J: ~ a (and Vo ~ B), the disorder is such that Ak ~ k (following Ioffe and Regel), and under such conditions, at each collision, k randomly varies by Ak, the closest neighbour to k. In going from one potential well to the next, the wavefunction as detailed in eqn (23) randomly changes and , following Mott, looses its phase memory and should therefore be rewritten using an approximate form: \h(r) = LAn"'o(r - rs) . n

-

with An = Cn exp(i I ((VoIB) just above single order value): initial delocalisation and medium disorder In a system which corresponds to a great increase in disorder, and for the model of just two wells would correspond to an increase in the depth between the wells as in IVI - Vz] = Vo, the difference in energy, lEI - Ezl, increases to a corresponding level and AI differs from Az. The amplitudes of the functions are no longer constant and the wavefunction displays increasing disorder both in amplitude and in phase (Figure II-12-b).

II Electron and band structure

53

~o/~ , (a)

VriB'"I and . '" a; orbital j ust delocali ed

l i\

I I

(e)

I / I

...

/

!i \\1",12 I \ ~ .

' -.; ''''

...

Vo»B

Figure 11-12. Variations in wavefunction with delocalisation: (a) delocalisation-Iocalisation only; (b) weak delocalisation; and (c) strong delocalisation.

-

System in which (Vo/B) » I« Vo / B) well above single order value); strong delocalisation and great disorder

In this system a high ly locali sed state is formed, as shown in Figure II- 12-c, and as Vo increases the localisation is accentuated. In addi tion, there is no longer propagation along a line of potential well s and states are thus localised. An exponential decrease in the wavefunction starts to appear and is increasingly noticeable with increasing values of Vo. The wavefunction takes on the form

and can be rewritten

in which

~

is the localisation length.

54

Optoelectronics of molecules and polymers

To conclude, the factor Vo/B is a crucial term in deciding whether only localised states form (Vo/B > 1) or whether both localised and delocalised states can co-exist (Vo/B S 1). y-Band scheme andform ofthe states density function N(E) From a realistic scheme of the distribution of potential wells, we can see that states should be localised within one energy domain and delocalised in another. Accordingly, in Figure 11-13 is described a system with non-negligible disorder: -

-

all states at the tail end of the function N(E) which correspond to a high enough value of Vo and from energies E(EcandE)E~ appear localised as before in the scheme of potential wells ; however, the middle of the band corresponds to shallow states with small Vo, such as Vo/B, and is a zone of delocalised states which have E~ < E < Ec.

d Localised states, conductivity and Anderson's metal-insulator transition a-Molt's definition : Mott's definition is based on continuous conductivity relative to electrons with a given energy (crE(O» and delocalised states are on average , at T = 0 K, those for which crE(O) is zero i.e. (crE (O») = O. To arrive at an average though, all possible configurations which have the energy E need to be con sidered, and while some electrons may have a non-zero energy, the average over all possible states with the corresponding energy E gives zero as a result. These states and the mobility they represent are in effect thermally activated. However, at T = 0 K, delocalised states average to give crE(O) =ft 0, that is to say metallic behaviour occurs . ~-State properties In Figure 11-13, two types of states-localised and delocalised-are separated by energies E c and E~ which together are called the ' mobility

v

v.{----E

Figure 11-13. Representation of localisedand delocalised states co-existence.

II Electron and band structure

Insulator or se mico nd ucto r character depends on po sition o f E f

55

Position of Ef result s in 'metallic' cha racte r

Figure11-14. Metallic character resulting from the domain EF.

edge ' . In the two zones Einstein's relation holds true if EF is outside the bands of non-degenerate states. This gives lJ., = qD /kT, but the diffu sion coefficients (D) have different forms , as in D = Pa 2 in which P repre sents the probability of movement to neighbouring sites . This brings us to the origin of the expressions used in Chapter 5: •

•

when E > E~ and E < Ec, D = (I / 6)Vpha2 exp( -wIlkT) (with Vph being the phonon frequency and WI the energy of activation) and (<JE (O) h =OK = O. Here as T ~ 0, we can verify that D and lJ., tend towards zero, much as conductivity; and when Ec < E < E~ , D = (l /6)v ea 2 and <JE(O) f= 0 where V e is the frequency of electronic vibrations.

y-Slightly disordered media, in which localisation is slight and E is small, and the distinction between an insulator or semiconductor and a metal As in the case of classic, crystalline media, the position of EF, as detailed in Figure 11-14, is related to the nature of a material. When EF is situated in the domain of delocalised states (E, < EF < E~) there is degeneration appropriate for a 'metallic' character. However, when EF is situated in the zone of localised states, for which typic ally E < Ec , charge carriers can only be thermally excited and conductivity can occur only by jumps or by excitation to Ec, and indeed at 0 K conductivity tends towards 0 which is typical of an insulator. Materials for which the Fermi level is situated in an energ y zone in which states are localised are called Fermi glasses. S-Metal-insulator or semiconductor transition For a given material which has a Fermi level fixed by its charge density, displacement ofEc, for example by increasing the disorder as shown in Figure II-IS , moves the Fermi level from an initial state in a domain of delocalised state s (metallic) to a zone of localised states. The result is a metal to insulator or semiconductor transition . e-Anderson transition from order to disorder and the change in conductivity Even though we do not detail transport properties in this Chapter, we should nevertheless introduce an expression for metallic conductivity written in the relatively simple form of <J = qnu = nq21: /m*, in which n is electron concentration and 1: is the relaxation time with respect to the Fermi level. With { = V1: we have <J = nq2{ /m *v, and on introducing the crystalline mornentum.h k = rrr 'v, we reach <J = nq2 £ /ft kF in which kF is the wave vector at the Fermi surface. We can also note that the number n of elec trons within a unit volume V(V = I) can be obtained by use of the reciprocal space,

56

Optoelectronics of molecules and polymers (E)

Changing level of orde r ean di place Ec within this zone

E

Ul

Figure 11-15.Using disorder to displace Et and effect metal-insulator transition.

that is to say the number of cells within the Fermi volume being ([4/3]rrk{)/8rr 3 , each with volume 8rr 3 /V = 8rr 3 for V = 1. In taking into account electron spins (i.e. doubly occupied cells), we have n = 2([4 /3]rrk{)/8rr 3 and metallic conductivity can therefore be written as CYB = 4rrkF 2q2L:j12rr 3ft (cf Section III-I in Chapter V). When considering a metallic state, the Fermi level can be considered more or less atthe band middle, as in I-D with kF ex tc ]« (p. 21 of [Mot 93]) and Figure 11-14. With increasing disorder, Ec and E~ tend towards each other at the band centre EM(~ EF) at which point all states are delocalised. This change is called Anderson's transition and is detailed in Figure 11-16. Simultaneously, Vo/B ::: 1 with E tending towards a. For its part, with £ = a, conductivity CY tends towards CYmin = CYIR = (CYB) £=a = q2/3a/i. In mono-dimensional media this abrupt transition is a point of controversy

as it is known to occur progressively in 3-dimensions. When a is of the order of 3 A, CYIR = 700 S em -I, often a saturation value for conductivity in rising temperatures.

In 0

T sO

T

11 =0 0 =0

= 0;

1I ;t 0; o;t 0 (metal)

3· 0; sca le law

Fermi glass

+ --ir.'--

tran ition metal to insulator (8

or

+----

Increasin g di order

Ec

0)

-

Figure 11-16. Anderson's transition from metal to insulator at absolute zero following I /(Vo /B) = B /Vo . The same phenomena occurs as EF is displaced from Ec .

II Electron and band structure

57

V Conclusion The origin of energy bands in a perfect three dimensional crystal, a material which presents a perfect regularity tied to a geometric structure unaltered by any physical reality was presented in this Chapter. In addition , supplementary effects such as dangling bonds, chain ends and holes within the structure were considered. These imperfections introduced into the band gap localised levels which once fluctuated could open to form a band which could split as a function of electron filling, in a manner analogous to the perturbations caused by electron repulsions, which were not taken into account in the band theory. By introducing modifications of crystal regularity by considering network thermal vibrations (phonons) and defects, both chemical (impurities) and physical (dislocations), the notion of a real crystal was studied . This resulted in determining the free mean pathway of electrons, which could no longer be considered as completely delocalised within the network-as was the case in a perfect crystal. It was shown that an increase in disorder reduced the free mean path length up to the point of localising

~I ( E )

(e)

E ~(E)

mobility gap

T =OK

(b)

E (E)

Conduction band ,;;;;~ ( BC) of delocalised state '"///////,

(a)

"///////~

E ' (E)

(d)

Ev

E8 EF

EA

Ec

E

Figure 11-17. Band model s for amorphous semiconductors: (a) following CFO; (b) following /leE) at T = 0 K; (c) following It (E) at T > 0 K; and (d) following that of Mott and Davis.

58

Optoelectronics of molecules and polymers

electrons in neighbourhoods of deep defaults, resulting in energy levels localised at extremities, or "tails", of permitted bands. Finally, it can be noted that all models postulate for amorphous media such as crystalline semiconductors that there are conduction and valence bands which are or are not separated by a band gap, depending on the all important band tails. And that: -

-

-

bands result in part from short range order (from approximations of strong bonds giving rise to bonding and anti-bonding states, i.e. valence and conduction bands separated by a band gap) and from disorder created by phonons or impurities shown by tails of delocalised states . Tail states are neutral when occupied in the case of the valence band and when empty in the conduction band. The Fermi level is thus placed in the middle of the band gap, as shown in Figure II-17 and following the model proposed by Cohen , Fritzsche and Ovshinsky (CFO). the form of the bands depends on the type of the implicated orbitals . For p or d orbitals, which are less stretched overall into space than s-orbitals , the form of N(E) is different and the bands are smaller. in a perfect crystal , the band gap is an forbidden energy in which N(E) = 0, while in an amorphous material it is a mobility gap and N(E) is not necessarily zero but the mobility !-L(E) however does becomes zero at T = OK (localised states), as shown in Figure 11-17-b and c.

By taking into account the disorder caused by not only phonons and impurities but also by structural defects such as dangling bonds and chain ends, additional offsetting defaults localised in the middle of the band can generate two bands at compensating levels (Hubbard's bands) following the model of Mott and Davis as shown in Figure II-17-d .

III Electron and band structures of 'perfect' organic solids

I Introduction: organic solids 1 Context Here the construction of a band scheme, including permitted and forbidden band s, will be detailed for ' perfect' organic solids. ' Perfect' means a neutr al solid which has no excess charges and no default s due to impurities, electronic structural default s such as topological faults associated with quasi-particles or solitons, and no geometrical faults due to dangling bonds or disorder. The latter, however, will be evoked. All these 'contributio ns' will be detailed in Chapter 4. Coming back to the present Chapter, we will limit ourselves to establishing a band scheme model normally used for physical studies of intrinsically, semiconducting inorga nic solids without defaults. Network distortion, due to the stabilisation of the electronic structure of polyace tylene, based on 'perfectly' alternating single and double bond s of fixed lengths, will however be considered.

2 Generalities In preparin g optoelectronic comp onents, we need to use materials which go beyond being passive such as the organic solids used as insulators. The latter materials , for example polyethylene based on the repeat unit -(CH2)n- are dielectric and have very high forbidden band s of at least 5 eV, an energy level situated well outside the optical spectrum, disfavourin g electronic transport. Thi s results from the considerable energ y separation between molecul ar bondin g (a) and anti-bonding (o") orbitals joining CH2 group s, which is in turn due to con siderable axia l overlapping of these orbitals allowed by the polymer geo metry [Ngu 94] (see also App endix A- I) . We should realise, however, that energy separation between bonding 11 and anti-bonding 11* -orbitals is relatively small, as lateral orbitals display limited overlapping. The band gap for molecular or polymeri c solids co ntaining such orbitals is typically between I and 3 eV, a value which perm its the use of their optical and electronic transport properties in the domain of optoelectro nics.

60

Optoelectronics of molecules and polymers

""

/

(a)

(b)

Figure III-I. (a) and (b) Energetically equivalent formsof polyacetylene (-(CHh-).

Here we will look at organic materials based on polymers displaying It-conjugation and small molecules containing It-bonds. Each material type presents its own advantages and disadvantages. Polymers often display high thermal stabilities and are generally considered well apt to forming thin films over large surfaces, for example by spin coating. However, this process does require the use of solvents making preparations of films containing more than one coat difficult. In this Chapter we will look at two types of electronic structure of It-conjugated polymers using as examples polyacetylene (Figure III-I) of which the fundamental energy state is degenerate due to two possible configurations, as represented in Figure III-2-a, and poly(para-phenylene) (PPP) which is based on the repeat unit structure -(C6H4)n- and, as detailed in Figure III-3-a, has a nondegenerate fundamental energy level. We should also note that the widely used It-conjugated poly(para-phenylene vinylene) (PPV), represented in Figure III-3-b, is also non-degenerate. Materials based on small molecules, however, require evaporation in vacuum chambers and delicate, although now well controlled, handling. New alternative deposition technologies have been developed using ink-jet or roll-to-roll printing. These materials are generally easy to purify, thus reducing reactions and diffusions at electrodes , and can sometimes be better ordered than polymer based materials even to the point of displaying higher charge mobilities, as in the case of crystallised small

Energetically equivalent forms of (CH)x

Non-equivalent benzenoid and tUinoid forms.

«» (a)

(b)

Configuration

-----co--·~r·dinates

Figure 111-2. (a) Degenerate (CHh ; and (b) non-degenerate PPP.

n (a)

(b)

Figure III-3. Structure of: (a) PPP; and (b) PPy.

n

III Electron and band structures of 'perfect' organic solids

61

molecules. We will limit ourselves to looking in a more or less qualitative manner at the band structures of two molecules which are widely used:

o 8-Tris-hydroxyquinoline aluminium (Alq3) represented in Figure I1I-4 is an

organometallic complex based on a central metal cation co-ordinated with quinolate ligands . To render this material usable in electroluminescence based applications it is prepared as a thin film by evaporation under vacuum, which in itself demands that the complexes have no overall charge and saturated co-ordination numbers [Miy 97]. Alq3 satisfies this and is the most generally used complex, even if its fluorescent quantum yield is relatively low. It provides films exhibiting good stabilities in electroluminescence and is a good transporter of electrons. o Fullerene-60 (C60) or buckminster fullerene, represented in Figure I1I-5, is original in that displays a spherical distribution of n-electrons and is constituted of 20 hexagons and 12 pentagons, resulting in each carbon atom being in the same environment and having the same sp2 hybridisation state (modified by the spherical shape).

99 "i '& 7 /

~

::-,.

.

N

o

.

"AI"-O~ "'. '? 0 '. I • N ,. I ~ I

N •••• •• /

::::-,.

I

s,

Figure 111·4. Structure of Alq3.

Figure 111-5. Representation of fullerene-60 showing bond co-ordination.

62

Optoelectronics of molecule s and polymers

3 Definition of conjugated materials [Ngu 94]; an aide-memoire for physicians and electricians In the simplest of terms , a conjugated system is one which has alternating single and double bonds . Examples of butadiene and benzene are shown in Figure III-6-a, which has been simplified by excluding bonds to hydrogen atoms. In Figure III-6-a we can see that each conjugated carbon atom has 3 neighbours, with which it forms 3 equivalent a -bonds resulting from the triangular sp2 hybridisation on atomic orbitals, which in this example are 2s, 2px, and 2py, cf Appendix A-I, Section II-I-b. The 4 th valence orbital of carbon , 2pz, which is perpendicular to the plane of a-bonds, undergoes lateral overlapping with other carbon 2pz orbital s to form norbitals. In fact, as shown in Figure III-7-a, the overlap between carbon atoms C \ and C2, and between C3 and C4 is dominant, locali sing double bond s, however, the overlap between C2 and C3 is not non-negligible and will lead to a better definition of a conjugated system with laterally overlapping p orbitals . In Figure-III-7-a there is only I orbital at C2 to assure bonds with neighbouring atoms C\ and C3. There should be 2 orbitals but as there is only I we can conclude that there are not enough orbitals or electrons to assure the presence of 2 saturated n -bonds, which require 2 electrons per bond . Similarly, we can see that in the case of graphite shown in Figure III-7-b that at C2 only a single orbital is present to assure bonds with 3 neighbouring atoms (C\ , C3, C4). There are therefore not enough orbitals, as 3 are required, nor electrons, as again 3 are required but only I is available, to assure the

Benzene

Butadiene Cz

~C4 C3

C1

(a)

0

(b) H

(a -bond between C 1 and

2)

Figure 111-6. (a) example of conjugated structures ; (b) hybrid a-bond between C\ and C2.

(a: butadie ne)

(b: graphi te)

Figure 111-7. a- and It-bonds in: (a) butadiene ; and (b) graphite (b).

III Electron and band structures of 'perfect' organic solids

63

presence of 3 saturated n-bonds. It is this non-saturat ed charac ter of the delocalised bonds which gives rise to the conductivity of graphite, [Moo 93 ] assoc iated with an electron cloud between each plane of reticulated atoms. There are two comments which we can make at this point: (i) a delocalised bond can be defined as one which is common to three or more atoms and displays an equal distribution of electrons; and (ii) qualitatively noting, lateral overlaps, giving rise to rt-bonds as shown in Figure III-7 (ef Appendix A- I, Section II), are weaker than axial overlaps which result in the in a -bonds shown in Figure III-6-b. As previously described , the energetic separation between a n- bonding orbital and its corresponding n "-anti-bonding orbital is less than that between a- and o"-orbitals.

II Electronic structure of organic intrinsic solids: n-conjugated polymers 1 Degenerate Jt-conjugated polymers: polyacetylene, the archetypal 'conducting polymer' Polyacetylene was the starting point for research into conducting polymers developed at the end of the 1970s [Sko 86].

a Delocalised n-bond structure with unit repetition constant a Generally speaking, when constructing molecular orbitals, as detailed in Append ix A- I, Section II-I , 2s 22p2 carbon states pass via an excited state, 2s 2px 2py 2pz. sp2 Hybridisation similarly consists in mixing I s-state and 2 p-states, as in our example with the configuration 2s 2px 2py, resulting in 3 equivalent hybrid orbitals, while having left the 4th orbital (2pz) unchan ged . The latter is represented in Figure III-8-a by black spots. We can consider CH units as being linked by a -bonds formed from a triangul ar sp2 hybridi sation and repre sented in Figure III-6-b. a-Bonds, which assure the structural rigidit y of the polymer, give rise to bands of a- bonding and o" -anti-bonding orbitals, which are separated by an appropriately large band gap-given the importance of hybrid orbital interactions. The representation used in Chapter II , Section III for sp3 orbitals can be used again here in a similar manner for sp2 orbitals: the a-bonding orbit als are completely filled by 3N sp2 electrons while o"-orbit als are completely empty, so that the electron transport assoc iated with these bands is zero. As represented in Figure III-8-a, given the geometry of polyacetylene , we can assume that the backbone is one-dimensional relative to 2pz electrons with unit repetition a. As detailed in Figure III-8-b, with N carbon atoms in a chain and N pz electrons which fill only half of the first band (which can contain 2N electrons as shown in Chapter I, Sectio n VI-2). polyacety lene should behave as a metal (half-filled

64

Optoelectronics of moleculesand polymers E

(b)

(a)

Figure 111-8. (a) configuration of polyacetylene with unit repetition constant a; (b) resulting band scheme, as classically shown, corresponding to a weak bond approximation, however, consideringa strong bond approximation yields a similar representation, as shown in Figure1-10 [MolSS] .

last band), however, this is not the case. In order to take into account reality, we need to bring in Peierls distortion, which results in a dimerisation of the chain.

b Conjugated structure with localised, alternating single and double bonds and repeat unit 2a Peierls distortion, in terms of energy, favours dimerised structures, such as shown in Figure III-9-a, in I-D. As we have seen in Chapter I, Sections V-2 and VI-3, the energy of deformation (Edefor) is less than the gain in electron energy following the opening of a band gap (ErelaxJ . The unchanged o-orbitals maintain polymer rigidity, however, the permitted band can only take on N electrons, as its size has been halved (Figure III-9-b). Therefore, N pz electrons from N carbon atoms remain blocked within this band. We can see that polyacetylene in its natural state is a semiconductor, with a forbidden band gap of the order of 1.5 eV. Levels associated with defaults (solitons) and electronic doping will be detailed in Chapter IV.

H

r

2a

H

I

/Cy/C~~(--C H

/

I

H

H (a)

~ Er
H

H

~I

I

c

..._.

C

I H

I

I k _ 2a .!. r It It

-kr (b)

Figure 111-9. (a) configuration of polyacetylene with period 2a; (b) band scheme for configuration (a) which can be compared with the representation shown in Figure 1-16 of strong bonds.

III Electron and band structures of 'perfect' organic solids

65

2 Band scheme for a non-degenerate n-conjugated polymer: poly(para-phenylene) a Basics Here we shall treat PPP, based achain of benzene rings (-(C6H4)n-), in the framework of Hiickel theory. We will initially establish the form ofthe band scheme for an isolated polymer. We will then use techniques previously used for strong bonds, i.e. covalent, intramolecular bonds, which are appropriate for organic materials. Intermolecular interactions will not be con sidered in the initial stages. As mentioned above, overlapping being weaker between rr-orbitals than between o-orbitals, the n-n * separation is weaker than (J'-(J'* separation. We will thus be limited to studying band s resulting from n- and n *-molecular orbitals, which correspond to highest molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) bands. The se bands , which have an energy interval in which are distributed bonding and anti-bonding state s, are thu s analogues of valence (VB - the last full band) and conduction (CB - the first empty or partially occupied band) bands , which are traditionally introduced in solid state physics in a band scheme of weak bonds . We shall consider o-bonds as contributing only a constant bonding force between atoms. We shall thus obtain a definitive band scheme in three stage s: -

-

-

in the first stage, we shall determine as simpl y as possible the energy states of an isolated benzene ring. Thi s can be done by treating the problem in I-D using results obtained in Chapter I with Floquer 's theorem. Appendix A-2 , Section II also gives relevant descriptions of n -p and n "-p orbitals; secondly, we shall con sider state interactions of a benzene ring within a polymer chain. This will result in a breakdown of nand rt " band s, in a mechanism identical to that detailed in Chapter II, Section III-3 for molecular orbital coupling; finally, we shall involve interchain interactions by cons idering the behaviour of amorphous semiconductors (Chapter II, Section IV), to then propose a definitive band scheme.

b Energy states and orbitals of an isolated benzene ring Molecular orbitals of an isolated ring , also considered as a cyclic conjugated polyene, can be obtained by considering linear combinations of atomic orbitals, with FIoquet's theorem as detailed by eqn (10) in Chapter I. For a regular system , of repeat unit length d, molecular orbitals are in the form \}Ik(r) = Co L s eik .rs\}Is(r ) in which functions \}I,(r) are pz type atomi c orbitals from 6 benzene carbon atoms each designated by its s number over which the summation is performed, and k is determined with the help of Born von Karman cyclic conditions which state that \}Ik(r) = \}IkCr + L) , in which L = Nd and being the regular chain length , here containing N = 6 bonds of length d. Using e ikL = I give s k in term s of k p = 2n Jd = n:fcr with p varying in integers from -3 to +3 corresponding to value s in k from between and +£ in the first zone .

-J

66

Optoelectronics of molecules and polymers

Six functions , \Ilk (r) = \IIkp (r), associated with 6 energy levels in the form Ep = E~ + 2~o cos(kpd), correspond to 6 possible values of p and are represented in Figure III-tO, in which are also shown wavefunctions and energy levels for an isolated benzene ring. Here we should note that: • wavefunctions can be progressive, with one half having propagation in sense '!rkp with the other half in the opposing sense '!rL p , or stationary, resulting directly from real and imaginary parts of wavefunctions given by Floquet's theorem (an example is given in Chapter I, Section VII-3). For the energy level E p = ±I, stationary solutions result in antinodal (real part in cosine) or nodal (imaginary part in sine) solutions in para positions ; • E~ is the Coulombic integral representing energy of a 2pz electron on a carbon atom, while ~o is the transfer integral, also known as exchange or resonance integral, between two adjacent carbon atoms in the ring; and • taking spin into account, the energetic separation between HOMO and LUMO is equal to 2~o (empirically determined to be ca. 5.5 eV) and the three lowest levels are occupied.

:

I I I I I I I I I

:

Antinodal I

p = 3. 'IlkJ

'1\.2 P = ± 2 --"":',,:f"k2

t I

: 2 ~o = 5 .5 e V

Nodal ~oIUli~

:

solutions in

I

+ E\ In parlllfli\.\ p = ± I

position

p =O. 'I'

tI

--rrBEL ZE IE

/

I

1"'''', ilion

I

i '---I I I ~ I

i I I I I I I I I I

benzene internal rcsnnance lntegrnl

Conduction band n'

I

EncrJ::Y levels given by COlli on 's cos t~n 13)1

~o:

Encrgy

Valence band n

l'ol)°(p-phen)lenc) (1'1'1')

~o

II.: rcsonance integral between phenylene rings

Figure 111-10. Energy levels for: left, benzene ; and right, PPP.

III Electron and band structures of 'perfect' organic solids

67

c Coupling of orbitals on adjacent polymer rings-generation of It- and It*-bands

If we now consider a polymer chain using the inter-ring transfer integral ~I which designates the coupling of adjacent, para positioned rings, we have a breakdown of the band of HOMOs and antinodal type LUMOs giving rise to Jt and Jt* bands, each of size 4~ I . Evolution of the band scheme can be observed in going from left to right of Figure III-10, and we can also note that energy levels corresponding to nodal solutions for energy levels E, and E2 remain, however, discreet. Variations in the state density function are shown on the right of Figure III-IO, in which we can see an appearance of a clear contribution due to breakdown of rt and n "-bands (leading eventually to discreet levels corresponding to nodal solutions when stationary solutions are privileged). While the band gap size of PPP has been experimentally evaluated at 3 eV, the relationship EG = 2~o - 4~ I can also used, given that we can estimate the size of n- and Jt*-bands at 4~1 = 2.5 eV. These values are well within the order of those determined by electron energy loss spectroscopy. While this is a rather simplified description of energy levels in PPP, rigorously detailed descriptions can be found elsewhere such as that by J.L. Bredas in [Sko 86] and references therein.

d Effects due to inter-chain interaction s and disorder

We should highlight the fact that the results obtained up to now have only dealt with intra-chain interactions in mono-dimensional models, and that inter-chain interactions, which appear once we reason in 3-D, will modify the results especially at the level of transport processes [Zup 93]. Inter-chain transfer integrals of around 0.1eV have been obtained for PPV [Gorn 93], a polymer with a similar conjugation to that of PPP. We can consider that inter-chain mobilities, which clearly determine to a high degree the total resulting mobility, are thus very much lower. In addition, a sample of PPP normally contains polymers with different lengths of conjugation, and with increasing chain length there is an increase in the distribution size which Jt and n " states, situated at band edges, undergo, resulting in bathochromic effects. Of one thing though we can be certain: in using results obtained for amorphous semiconductors, [Sko 86 and Mol 981 these effects will remove the brutal discontinuity observed for an isolated chain at the band edges; instead we will now obtain band tails, resembling those of amorphous semiconductors such as silicon. In addition, as in the case of carbon detailed in Chapter II, Section IV-I, forbidden states in the middle of the band can be added to allow for dangling groups, which can appear especially during thermal (e.g. in synthetic procedure) or radiation treatments. We will add these levels to the definitive band scheme after considering electron doping of PPP in Chapter IV.

68

Optoelectronics of molecules and polymers Energy

a· -anti bonding band

It*-antibonding band

. .::0....:------

It*-band tail

. .- - - - - - - - It-band tail It-bonding band

.-t-- -

a-bonding band

Figure 111-11.Band scheme for a rt-conjugated polymer.

III Electronic structure of organic intrinsic solids: small molecules 1 Evolution of energy levels in going from an isolated chain to a system of solid state condensed molecules In polymers, each monomeric unit is joined to its neighbours by strong bonds with transfer integrals ~l which lead to the breakdown of bands (Figure III-lO). However, in thin films of small molecules, cohesion is due only to weak Van der Waals forces. The resulting coupling between molecular orbitals is weak due to poor overlapping between orbitals of the molecules in the material. In addition, the greater the distances between molecules, the weaker the interactions between electrons on two neighbouring molecules and the lesser the degeneration observed in going from an isolated to condensed state by valence electrons. The result is that the bands obtained for molecular solids are not very interactive and are narrow. In a general, qualitative manner, we can nevertheless indicate the way in which a band gap evolves with respect to the size of a total system of molecules. Note that the term ' small molecules' encompasses benzene, anthracene (3 joined phenyl rings) and even tetracene (4 joined phenyl rings) and the discrete occupied levels increase to such

III Electron and band structures of 'perfect' organic solids

69

a point that we have a considerable number of energy states for the HOMO [Pop 82]. Finally, the separation, or band gap, between HOMO and LUMO bands decreases with increasing numbers of rings, just as for polymers in which energy levels become more and more 'dense' as we increase chain lengths . It should be noted that the term 'band' here corresponds simply to an energy interval in which HOMO and LUMO levels are situated, without making any strict reference to a Bloch pseudo-continuum of energy levels. It is interesting to remember though that the state, whether neutral or charged, of solid state molecules equally effects the band gap size ; polarisation of the solid by ionised molecules actually decreases the band gap [Pop 82]. Without going into some tedious and highly specialised calculations, we shall limit ourselves to simply indicating the electron band structure for two types of molecular solids : -

Alq3 which is pretty much amorphous and is widely used in the field of electroluminescence; and C60 (in the undoped state is called fullerite and in the doped state is called fulleride) crystallises into a cubic face centred (cfc) pattern and shows great potential for use in photovoltaic systems .

2 Energy level distribution in Alq3 a Generalities Optical spectra of Alq3 evaporated under vacuum into a thin film (solid state) or in solution (diluted in DMF) have been empirically determined to be more or less identical. This has shown that intermolecular interactions are negligible in the solid state and any corresponding breakdown of bands is reduced. If the inverse were true, then we would observe strong intermolecular interactions and a high degree of delocalisation of charge carriers throughout the medium, with for example electrons or holes readily moving from one molecule to the next giving rise to an increased charge mobility and a consequential increase in the size of permitted bands . As we are looking at Alq3 , a large molecule consisting of a central metal cation tied to 3 surrounding ligands, we shall have to consider internal interactions, including those between ligands . A study has been made using semi-empirical determinations of molecular orbitals and associated energy levels giving a method intermediate between the Huckel and ab initio methods , which are normally reserved for small systems serving as ' reference models ' [Riv 89].

b Quantitative results from Alq3 proposed by Burrows et al In considering an isolated molecule of Alq3 and assuming it exhibited equal characteristics to a molecule within a thin film, by ignoring external effects due to weak Van der Waals forces as noted above, energy levels were determined by Burrows et at [Bur 96]. HOMO and LUMO band energies were estimated using the semi-empirical ZINDO method, which necessitated the use of various molecular configurations. Calculations indicated that there should be 3 small optical transitions corresponding to

70

Optoelectronics of molecules and polymers

Phcnoxidc group

(1I0MO)

Figure III-12. Localisation of full (HOMO) and empty (LUMO) orbitals on Alq3.

wavelengths A. = 377, 369 and 362 nm equal, respectively, to energies of 3.28, 3.35 and 3.42 eV, and the result was close to that experimentally observed at 385 nm. It was shown in the calculation that two isomers of Alq3 gave very similar results, indicating that solid state thin films of Alq3 are stable, undergoing negligible recrystallisation, and contain both isomers. Calculations indicated, amongst other things, that n- and rr"-orbitals were localised about quinolate ligand s, shown in Figure III-12 , and more specifically, full n-orbitals (HOMO) were situated at the ligand phenoxide group s while empty Jt * -orbitals (LUMO) were about ligand pyridine groups. The model has been tested and confirmed correct by various studie s in which substituents have been placed on the ligands , for example electron accepting groups once placed on the phenoxide group resulted in a bluer emission, in agreement with expected reductions in energy of the highest occupied level.

3 Fullerene electronic levels and states a Structure of C60 The spherical form of C60 results from the interstitial placement of 12 pentagons, which never touch, and 20 hexagons, as presented in Figure III-5 [Had 86]. Pentagon sides correspond to simple valence bonds which share from each carbon atom one electron and have length 1.45 A, while each pentagon is joined via double bonds which share from each carbon atom 2 electrons and have length 1.40A; each carbon atom is thus shared between one pentagon and two hexagons . Overall , C60has strong rotational symmetry consisting of an icosahedral (20 faces) (Ih) with six 5 th order rotational axis , ten 3rd order axes, fifteen 2nd order axes and one symmetry of inversion .

III Electron and band structures of 'perfect' organic solids

71

The full common name of C60 comes from that of American architect Buckminster Fuller who designed similarly structured dome-like building s. Double bonds within the structure of C60 result in two particular effects : •

•

conjugation between pentagons of delocali sed n-electrons, resulting in reduced electron-electron repulsions and increa sed polarisability. The system is highly stable and has an elevated electron affinity, estimated by photoelectron spectroscopy to be around 2.5 to 2.8 eY. Such a value indicate s that C60 should be a good electron conductor; even considering the non-planar structure, carbon atoms must be considered as undergoing a hybridisation which is part sp2 and part sp3 as Jt-electrons are not purely p type due to the poor pyramidal form of each carbon atom . Accordingly, we shall have to perform a slight 'rehybridisation' of the system by introducing some 2s character (estimated at around 10 %) into the Jt-orbital.

b Electronic levels of C60 calculated using Hiickel approximations A simple but approximate (given the last remark of the preceding paragraph) method to derive the electronic levels in C60 consists of considering its structure to be based on 3cr-bonds and IJt-bond per carbon atom. We can see this as an initial step in determining electronic levels of the 60Jt-electrons in the molecule (see also Appendix A-2, Section II). Hiickel 's method, briefly considered using a simple example in Appendix A-I , Section II, is based on a method of variations which establi shes t equations determining t energy levels (each of which may eventually be degenerate) associated with t electrons in the system. Here, t = I to 60. Thus we take following Hiickel's approximation (see Section IV-3-b): -a = Hlt and -~ = Hts Ilts, in which Ilts = I if s = t ± I or otherwi se Ilts = 0 which means that only resonance integrals for adjacent neighbours are taken to be non-zero ; and that overlap integrals between adjacent orbitals can also be consider negligible . Within the parameter -a, which has no effect on energy variations, we finally arrive at [Tro 96]: (Er)t=I .. .. 60 = -~

Matrix Of) closet ( neighbours

The matrix of closest neighbours is that in which the elements are (Ilts)t s with indices outside the brackets denoting the line and column of element Ilts = I , if t and s are first neighbours . A calculation of proper values of this matrix gives the energy scheme, graduated with units in ~ , shown in Figure 111-13. Names given to energy levels originate from corresponding molecular orbitals and their symmetries in group theory. There are 3 molecular orbitals at 0.139 ~ (level tl u ) and 3 more at 0.382 ~ (level tl g ) , so we can expect the molecule to have up to 12 electrons in these levels. The separation of hu and t I u levels can be estimated as t>E = O.76~, while that between hu and tl g is ~ , and is intermediate to that of anthracene (t>E = 0.83 ~) and tetracene (t>E = 0.59 ~) .

72 3

Optoelectronics of molecules and polymers Energy (in units of~) __ t

--

__ t

- - gu

lg

- - t lu

o ~ ~

-I

~ ~

:f =I -J,t.

-2

-3

g

ag

~ ~ L=O

]

~ hu

~ ~ tzu *

~

L~j

~ ] L=3

gu

*

hg

"* * L=2

~ t l U L=I

Figure 111-13. Energy levels associated with C60 Hiickel molecular orbitals

c Band scheme of solid

C60

The structure of C60 in the solid state resemb les that of a crystal and at ambient temperature centres of gravity of C60S occupy nodes of a face centred cubic (cfc) structure, however, as a function of time , each molecule turns about its own axis which is randomly and independently orientated with respect to other molecules in the lattice. A molecu le can thus be perceived by its neighbours more and more like a uniformly charged sphere. While stro ng covalent bonds assure the structure of each individual C60, only weak Van der Waals forces betwee n each molecule (Figure III-14) assure cohesion. With the overlap of orbitals being poor, the actual energy bands due to accum ulated C60 molecules remain narrow. The band derived from the HOMO, hu , is a valence band and is completely full, whereas the band derived from the LUMO lcvel.t nj , is a conduction band and is comp letely empty. Solid C60 (fullerite) is thus an insulator, or rather, a semico nductor as the band gap is estimated as being somew here betwee n 1.5 and 1.9 eV (Figure III- IS). Forma lly, a transition between these two bands, of the same u type, is forbidden, however, this selection rule is invalidated by electron-phonon coup ling, as has been obser ved (Chapter VIII). While the 'band structure' presented in the following Figure III-IS is qualitati ve, we should remember that in the solid state, molecu lar orbita ls remain narrow and tend towards those OfC60 in an isolated state. It has been observed [Kel92] that absorption spectra of films of C60 are very similar to those of molecul ar C60, underlining the weakness of intermo lecular interactions in the solid state. In addition, the similarity

III Electron and band structures of 'perfect' organic solids

73

Figure 111-14. Plan projection (001) of atom position in C60 in solid state structure cfc. Continuous lines, either thick and joining two hexagons as in Al A2 or thin and short as in A IA4, represent strong, intramolecular covalent bonds. Dotted lines represent weak, intermolecular Vander Waals.

Energy(p)

Conduction band (empty)

tl"

Forbidden band

o hu

Valence band Molecule

(full)

Solid

FigureIII-IS. Huckel approximation of the band scheme of solid state C60.

of spectra obtained for solid state and solvated C60 allowed attribution of solid state bands of energy levels associated with molecular orbitals. However, ongoing studies are being performed to determine the band structure of C60 in the solid state, their interest being to accurately describe the semiconducting properties of C60. The calculation methods used are either that of self-consistent field (SCF) or those based on so-called density functions, in particular the approximation

74

Optoelectronics of molecules and polymers

called the Local Density Approximation (LDA). A useful reference for the latter is [Tro 96], and Appendix A-3 details the principle results [Tro 92]. Here we will suffice ourselves with the three main results: •

• •

the band gap can be without an intermediate (at point X in Brillouin's zone, cutting the intersection of the axis kz with the upper face of the octahedral and defining Brillouin's zone of the cfc lattice); while the band gap has been found to be U8 eV [Tro 92], it is more generally estimated to be between 1.5 and 1.9 eV; and the greatest valence band size (in X) is between 0.42 eV and 0.58 e V.

IV Conclusion: energy levels and electron transport Most of the electronic energy level schemes in this Chapter have been prepared using conclusions from Chapter I . Within strong bond approximations, for a chain of N atoms, energies can be written as E p = -(X - 2~ cos(2npjN) in which p takes on N integer values such that -N j2 ~ P ~ Nj2, yielding N values for E. As shown in Figure 1-20, we now know that when N is low, successive p energy levels are well separated, and when N is high, then the spread between different levels is small. In the limiting case when N -+ 00, the cosine argument 8p = 2npjN results in a tendency towards a pseudo-continuity between -n and n; the electronic spectrum takes on the form Ep = -(X - 2~ cos 8p and the separation between two successive proper energy values tends towards zero, with a spectrum size which is thus proportional to ~ . The name 'energy band' is given to proper values of resulting spectra in reference to free or semi-free electrons (weak links f.E ~ 10-7 eV -+ 0 for delocalised states) . In the more general case in which a chain has a finite length, 8p shows discreet values along with the energy spectrum; a decreasing N results in an increase in the energy separation between two successive levels to the point at which a pseudo-continuous energy state spectrum can be transformed into a spectrum of discreet energy states. The latter can be observed more easily when it results from small molecules which display a succession of discreet energy states, which may only be slightly enlarged by weak intermolecular interactions (Van der Waals). However, with macromolecules in which N is large the only the result we can hope for is for an energy band assimilating energy state spectra. So, in a general and qualitative manner, we can retain from the examples studied that: •

•

long chains of degenerated polyacetylene with a high value ofN can be considered, using either weak or strong bond approximations, to have to all practical extents a continuous energy band. As we have seen in other cases the energy band is only half-filled with I rt-electron per carbon atom and the material should be a priori a conductor, however, a Peierls transition associated with the formation of alternating double and single bands explains its intrinsically insulating character; poly(para-phenylene), which is non-degenerate, typically displays a conjugation across 10 to 20 phenyl rings. Each ring results in discreet levels which are well

III Electron and band structures of 'perfect' organic solids

•

•

75

separated (note that for benzene, the band gap between HOMO and LUMO bands is ca. 5.5 eV). The coupling between adjacent phenyl rings though results in a breakdown of HOMO and LUMO JT- and JT* -bands and results in a band gap of the order of 3 eV between JT- and JT* -bands of size ca. 2.5 eV; Alq3 is an example of a 's mall molecule' , although relatively speaking it is not actually that small. Semi-empirical calculations (ZINDO) have been used to determine energy levels for an isolated molecule taking into account, at present, 15 configurations in which are included the 15 highest full sites and 15 lowe st empty sites. Calculations showed that phenoxide groups were the centre for HOMO levels, whereas pyridine groups had localised LUMO levels. In addition, weak Van der Waals intermolecular interactions do not result in even narrow molecular band breakdowns. Thi s has been indicated by the invariability of spectra of Alq3 in the solid and solvated state, showing no perceptible opening of discreet molecular bands levels. However, when considering electroluminescence, there is a problem in dealing with the continuity (or non-continuity) of energy levels susceptible to receiving injected charge carriers at interfaces. If the energy separation between levels is too great, then there is a risk that any injected charge will not find an accepting energy level. Given the size of the molecule though it is probable that there is a large number of energy levels and con sequently energy levels should be sufficiently close to ensure that charge carriers are not blocked. Th is problem has in fact given rise to some controversies and we will come back to this subject when looking at organic light emitting diodes (OLEDs) in a later Chapter; and in fullerene (C60) we have a molecule of a not inconsiderable size , thus discreet energy levels can be determined using classical Huckel theory for n-electron based systems.

In Figure Ill-15, and generalised in Figure 1lI- I 6, we see that in the solid state small molecules exhibit only a narrow breakdown of levels if intermolecular interactions are weak. As we shall see in Chapter V, charge transport mechanisms and optical Energy LUMO

HOMO Isolated molecule

2 Interacting molecules

Several interacting molecules in solid state

Figure 111-16. Band structure formation for a molecular system in the solid state in which HOMO and LUMO levels of individual molecules increase in size due to interactions with neighbouring molecules. More these interactions are weak, more narrow the bands become, reducing mobilities.

76

Optoelectronics of molecules and polymers

processes, which control opto-electronic properties, depend directly on the form and nature of energy bands/levels of a medium in the following way: •

•

if we have a 'pseudo-continuous' band structure, and permitted bands are sufficiently large, then conduction can easily occur between very close levels within bands. This, however, is rarely the case in organic solids ; if, however, we have a structure in which levels are discreetly spaced, then movement of charge carriers through different energy levels cannot occur without mechanisms allowing barrier crossing , and therefore carrier mobilities and conductivities are reduced . As we have seen before , this similarly applies to narrow permitted bands (see also Chapter II, Section II-2).

We should take note that band structures are actually considerably more complicated than that represented in Figure III-16. Supplementary levels are generated by a wide variety of cause s. For example we can construct: discreet levels resulting from dangling bonds; localised levels associated with structural, chemical and other forms of disorder; polaron levels due to lattice localised charges (with a polaron band displaying a specific form of transport) ; and levels due to traps, or deep wells. Before looking at transport properties in more depth, we shall examine the electronic structure of real organic solids in Chapter IV.

IV Electron and band structures of 'real' organic solids

I Introduction: 'real' organic solids In this Chapter we shall complete the band scheme for ' perfect' organic solids, obtained in Chapter III, by taking into account electronic contributions in 'rea l' organic solid materials, particularl y n -conjugated polymers, due to: •

• • •

excess charges either introduced on purpo se (doping) or not, including electronlattice coupling (vibrations) yielding qua si-particle s such as polarons or bipolarons; topological defaults yielding soliton s, including linking of chemical groups ; structural defaults such as dangling group s due to thermal or radiation treatment, using for example UV or ionic implantat ion; and geometric disorder and its effect on local charge s.

From the above, we shall be able to construct a band scheme which, although quite general in nature, will allow us in Chapter V to subscribe to each electronic level appropriate transport processes.

II Lattice-charge coupling-polarons, bipolarons, polaron-excitons, solitons and their associated energy levels-and n-conjugated polymer doping 1 Introduction In this Section we shall look at the effects of introducing excess charges into a distortable organic solid and discover resulting excited states associated with quasiparticles such as polarons, bipolarons, polaron -exciton s and solitons which appear depending on the nature of the material under study. We shall define them in

78

Optoelec tronics of molecules and polymers

more detail though much later on. Excess charges, either electrons or holes, can be introduced into materials via different routes : • •

• •

electrical doping by charge transfer from a donor type dopant (n-type) or acceptor (p-type) into an organic solid which is typically a n-conjugated polymer; electric field effect doping in which an organic semiconductor or MIS structure (Metal-Insulator-Semiconductor structure) is subject to positive or negative electrical polarisation due to influence from a grill doped with either n- or p-type dopants, respectively ; unipolar injection of charges into an organic solid using an electrode; and bipolar injection of opposed charges (electrons from cathode and holes from anode) which following migration through an organic solid can generate certain quasi-particles.

2 Polarons a The 'dielectric polaron' If we take an ionic crystal lattice, as shown in Figure IV-I-a, and then place an electron on an ion, as detailed by the black point in Figure IV-I-b, then we can see that surrounding ions undergo a force due to the additional electron, and this

(c)

(a)

G 0 8

0 8 0

0 8

(d)

1-)

'-'

/-,

/-,

1+)

1+)

'-' ,""

...../-,- ) ' -'

/-,.?'

8 E

,_....

1+)

00000

0

J.

(b)

00000

f '

/-,

1-)-+

~/ - , 1+ ,_ ) ....

'-'

o

0

o

o

0

o

/-,

1 -)

'r

Figure IV·I. (a) Lattice of ions in relaxed state; (b) repositioning of ions in directions of arrows following placement of electron at E in the lattice; (c) and (d) polaron formation in covalent materials going from a regular arrangement of rare gas atoms in (c) to a deformed atomic lattice after a hole has been placed at D.

IV Electron and bandstructures of 'real' organic solids

79

electron-lattice interaction results in a new positions for the ions, as shown in the same Figure by dotted lines. This displacement of ions always results in a reduction of electron energy, and also results in a potential well within which can be found the electron. If the well is deep enough then the electron will find itself in a tied state, incapable of moving to another site unless there is modification in the positions of neighbouring ions. We can label the electron 'self-trapped' and as such, it and its associated lattice deformation is termed a 'polaron' . The term originated from phenomena observed in polar materials, however, such quasi -particles can also occur in covalent materials.

b Molecular polarons Molecular polarons form in covalent materials in which the resulting distortion in confined to neighbouring atoms, which can subsequently form a chemical bond while the charge is trapped. A good example is Vk centres in alkali metal-halogen crystals, in which a hole trapped on a Cl" ion results in attracting a neighbouring Cl- ion to yield a 'molecular ion' of form Cll". Similar phenomena can occur in solid rare gases in which a trapped hole, detailed in Figure IV-l-c and d, and similarly in certain mineral glasses in which dangling bonds at a neutral site (D°) can result in a more favourable local rearrangement such that 2 D° -+ 0 + + 0- where 0+ and 0 - are the previously neutral dangling bonds (D°) which have, respectively, lost or gained an electron charge . We will now go on to detail further the origin of polarons in molecular crystals. c Small and large polarons

If the wavefunction associated with a self-trapped electron takes up a space equal to or smaller than the lattice constant, then the polaron is called a 'small polaron' and the deformation is localised only in the neighbourhood of the charge carrier. We shall see this sort of quasi-particle in covalent materials. However, a 'large polaron' , sometimes called a ' Frohlich polaron' , is one which forms in polar media in which Coulombic forces are involved and which polarises the crystal along the length of its action . We will not detail the latter polaron (which is covered elsewhere [Cox 87]), and shall essentially consider polaron s in molecular or macromolecular media.

3 Model of molecular crystals

a Holstein's model in equation form Here we shall use molecular crystals to obtain a simple, generalised model for small polarons (following Holstein's model [HoI 59]) . We shall consider just one excess electron placed in a regularly aligned but flexible lattice of molecules each of mass M. With each molecule g we associate a co-ordinate xg which represents the movement of the molecule under harmonic vibration of pulsation

Wo

with frequency Vo =

2~~'

80

Optoelectronics of molecules and polymers

We start with a single molecular site, excluding coupling effects with surrounding molecules, on which there is not yet an excess electron . While ~(r) represents the ~ ---+ potential energy of an oscillator, the vibration force is F(r) = -grad ~(r). When ro is in an equilibrium position, such that xg = r - ro represents the local deformation of a molecule , we have, using a Mac Laurin development,

~(r) = ~(ro + xg ) = ~(ro)

+ -I (a2~) -2

ar

2

x~.

ro

The first derivative is in effect zero as ~(ro) corresponds to a minimum energy position at equilibrium for r = roo We now have F(r)

and with the value of k =

= _ a~ = _ (a2~) 2

ar

(~) ro

ar

given in

x ro

~(r)

g

= -kx

gives

~(r)

g

=

~(ro) + 1kxi. With k

verifying the harmonic equation d2xg F(r) = M - 2 = -kx g , dt for which the solution xg = X cos wot necessitates the introduction of an actual

IK,

we finally obtain ~(r) = ~(ro) + 1 Mw Xi· pulsation Wo = Wo can be estimated for an elongation xg of the order of a, the lattice constant, as vibration energy, MW5a2, is of the same order as that of a bond energy EL in

6

!

the molecule and EL ~ I eV. In taking a ~ I Aand M = J(~-3 (hydrogen atom mass 10- 3 kg), we have

giving wo ~ 1014 rad s-l . If a = I nm though, we now have Wo ~ 10 13 rad S-1 and for heavier atoms, the frequencies are even lower. On taking into account the coupling between the vibrational movement of a molecule and that of its neighbours, which results in the transfer of vibrational energy throughout the lattice, we have to bring a phenomenon of frequency dispersion co = f(k) into play. The Hamiltonian corresponding to this molecular crystal , into which we have not yet inserted free charges is written [Emi 86] as:

in which h designates the closest neighbour, M is reduced mass and 6 Wb is the size of the band of optical phonons .

IV Electron and band structures of 'real' organic solids

81

When a supplementary and excess electron is introduced into the lattice we can take account of electron-lattice interactions by considering the excess carrier energy at a site in the lattice . We accept that the energy is a linear function of movements within the lattice, and the greater the induced movement the greater the absolute value of the electron-lattice energy and the more easily that the charge is self-trapped. This trapping is actually be greater than any coupling. For a carrier localised on a site g, we can write Eg = Eo - Lgf f(g ' - g)Xgf, in which f (g' - g) is a weighting factor which characterises electron-lattice interactions. In assuming that interactions are over short distances in using f(g' - g) = AOgfg, the previous relationship changes to E g = Eo - Axg , in which Eo is the energy of a nondistorted site, Eg is Eo = 0 plus a constant and A represents the electron-site coupling force, as in A = - E:g-=-~o = -grad E. In general , we can write Eg = E(x g ) = -Axg . Thus in the scenario we are considering, molecular deformations induced by the charge carrier are mostly localised around the carrier itself, and it is the presence of vibrational coupling between neighbouring molecules that distribute distortion effects beyond the occupied site. Figure IV-2, otherwise known as an Emin representation, presents a scheme of the distance from equilibrium of diatomic molecules-represented by vertical lines-in a linear molecular crystal about a site with an electron (black dot). The complete resolution of this problem will require the use of Hamiltonian operators (He) for strongly bonded electrons in a crystal of covalently bonded molecules, which will add a Hamiltonian for vibrational energies (Hd . Results relevant to short range electron-lattice interactions, which we will need to consider here are also discussed in Section 7.1 of [Mol 94].

b Limiting case of a single molecular site: polaron and bipolaron a Polarons and single electrons In this limiting case we will place I excess electron on a single site, the molecule of which undergoes a deformation x. The potential

Figure IV-2. Scheme of equilibrium separation distances for diatomic molecules in a linear chain, in which an electron (black dot) is placed at the centre of a molecule.

82

Optoelectronics of molecules and polymers

energy of the site can be expressed as:

x2

E = M w6 - + (Eo - Ax) = Bx 2 '-.-' 2 '-.-'

Ax in which Eo = 0

-

i

i

i

Vibration energy of molecular site

and

B=

2I Mwo2 ·

,--..L..-------,

Energy when there is no distortion

Energy of e1ectronlattice interaction (electron-vibration coupling)

E is optimised for a value x = Xo such that ( aaxE ) x=xo =0 '

andMw6xo-A=2xoB-A=0~xo=~=~, Mw o

2B

Following a deformation of the lattice Xo through vibration and electron-lattice interactions, the energy of electron-lattice interactions is accordingly reduced by Axo (-Axo written algebraically). The system though undergoes a distortion associated with a vibrational energy equal to BX6 which is such that BX6 = B (2~) Xo = ~Axo . Overall we can write that energy in the system is reduced by Ep for which: I

Ep = BX6 - Ax., =

-2 Axo =

A A2 - 4B = - 2Mw6'

Representations of these various energies are shown in Figure IV-3. Finally we can note that in polar media, the formation of a polaron is highly favoured as the energy 'cost' due to a distortion is less in absolute terms than the energy recovered through a reduction in charge-lattice interactions. The former is actually exactly one half that of the latter. ~ Bipolarons and two electrons Using the model for a single molecular site, we can go on to consider the deformation due to two electrons. System energies can be written:

x2

E =M w62-2Ax +U~ - (2A + C)x + Vo ?'------------------,

Vibrational energy

Electron-lattice coupling energy

Coulombic repulsion due to 2 electrons on the same site . We can suppose that V varies following V = Vo - Cx, in which Uo is the repulsion at zero deformation .

On replacing A in the preceding Section by 2A + C, the minimum E is now given by: X=Xl =

2A+C Mw 2

o

and

EBP = -

(2A

+ C)2 2

2Mw o

+ Vo.

This system in which there are two electrons localised on a single deformation is called a bipolaron and is stable if the energy required for its formation is less than

IV Electron and band stru cture s of ' real' organic solids

83

Bx'

Energy due to distortion

(a) Vibrational energy

~"'- Ax

<,

Recovered energy from system relaxation

<,

...... ..... " ' <,

X,,!

:

... ........... <,' ......

. (b) Energy of electronlattice interaction

x

- Ax"

Bx- -Ax ~.

Stabilised system as has reduced energy

-, Eo = BX;; Ax" = (-A')!2Mw'

xo ............ "",.;.

-"'x (c) Resulting energy.

Figu re IV-3. Different energy terms with re spect to lattice de formation by an excess electron.

twice the energy of 2 isolated polarons, i.e. when : (2A

+ C)2

A2

- - --;:-+Uo < - - 2Mw5 MW5 The above relationship can be true when A and C are of the same sign and when Uo is not too high. The deformation x I imposed by the two particles (here two electrons, but can also be holes) can be advantageous in overco ming moderate Coulombic repulsions .

4 Energy spectru m of small polaron As we have just seen, a supplementary electron which distor ts a molec ule onto whic h it is placed has an energy reduced by - Axo = 2E p , while the vibration energy of deformed molecule increases by BX5 = IEpl , resulting in an overa ll reduction in

84

Optoelectronics of molecules and polymers

Ep

with respect to the energy of an electron in a energy for the system equal to rigid crystal of molecules, for which xg = O. Within the limits of the preceding calculation, in which we have ignored vibrational dispersions due to coupl ing with non-exi sting neighbouring molecules, we have

IEpI =

A2

2Mw6

(see Figure IV-4).

If we also take into account the possibilit y that a small polaron (charge carrier and associated lattice deform ation ) can equally be situated at any other of the geometrical equivalents in a crystal, then we find that the actual states of the system are shared in a polaronic band as shown in Figure IV-4. Using a modified method of that used for strong bond appro ximations (eqn (2) and followin g eqns in Chapter II), we can consider that the proper states of a small polaron in a cubic crystal has energy in the form: Ek = -21 exp(-S)[cos kxa + cos kya + cos kza] - Ep ,

with :

Jk~ + k~ + k~ as the polaron

•

k=

•

1, the resonance integral between two 'electronically coupled' closest neighbours, is in the form exp (-aR) to take into account the exponential form of electronic wavefunctions ; and exp (-S) is an overlap factor associated with chain vibrations. It represents the superposition integral between two wavefunctions, which detail the vibrational

•

wavevector, and a the lattice constant;

Atomic energy level for an electron in a rigid crys tal

Bonding energy due to polaron formation with molecular deformation in presence of charge

Overlap integral for 2 wave functions denoting system vibrational state for a charge at one or other of two neighbouring sites

Figure IV-4. Energy scheme for small polaron.

IV Electron and band structures of 'real' organic solids

85

state of the system when the charge carrier is on one or another of two adjacent sites in a crystal. In the limiting case of a rigid lattice, the vibrational wave function remains unchanged during charge transfer from one site to its neighbour, and the overlap factor is equal to 1. However, here where we are looking at a distortable lattice, then exp( -S) can be considered to relate the neces sary overlapping of atomic sites between which the required tunnel effect can occur to allow a complete displacement of atomic site and charge, which is taken into account by J. Thinking about it in more depth, we can see that the transfer of a polaron requires in fact two , concomitant tunnelling effects. One is associated with moving a charge between two neighbouring sites (electronic resonance integral), while the other is concerned with displacing the deformation itself and any sites geometrically tied to the deformation. Exp( -S) is a factor of the same order as the atomic tunnelling effect as we can use it to assimilate atomic site transfers . As it is associated with the high mass of atoms, relative to the charge carriers, its value is extremely low. And thus for a cubic crystal, the polaronic band size is ~Ek = 12 J exp( -S) and is therefore extremely narrow, of width typically below that of vibrational energies, which are at least kT ~ 10- 4 eV at not too low temperatures.

Comment We need to remember that displaced deformation is equivalent to an 'exchange' in position for two neighbouring sites. Evidently, it is not two actual atoms which change place but their position relative to the deformation propagated with a charge. However, Figure IV-5 shows how geometrically speaking the exchange is equivalent to an interchange of 2 sites.

5 Polarons in rr-conjugated polymers Polarons in rr-conjugated polymers are of the same type as those we have looked at just above, although local reorganisation can be favoured by there being a stable ionic form of the polymer different to that associated with its neutral state. Here we shall

Chain at momentt

Vibration

1--+--.....- - - + - - - - - - - - Site (i+1) at instant t' plays the same role as site i at instantt;

propagation between instants t and t' '"

similarly i plays the role of i + I but atoms at site i are the same as those at site i' and remain as

interchange of i and i + I sites. i' + 1 behaves as i while i' behaves as i + 1

they are. as for atoms at sites + I and j' + l.

I----~----.--I------i

i'

i' + 1

Chain at moment t' = I + !'.T (i becomes i' and i + I becomes i' + I).

Figure IV-5. 'Interchange' of2 neighbouring sites during vibration (and polaron!) propagation, in which i and if are initial and final states.

86

Optoelectronics of molecules and polymers (a) Benzene ring form stable in neutral state

(c) loniscd stable quino id form: polaron formed

CB L 1\10: ami-bonding slalCs p doped

E

F

------------,

--t+---VB

~ ~ ---=-=-=-- - --=--:-::-: -- --(b)

(a)

(e)

Fundamerual 2 neutral state unstable CR dcfaults ndoped"

------

E

E

I charged default (polaron)

Heavily duped

F I~--~ - rrtftt ===e= 1

E

~· IP.~·IP:=tL VR

2 charged Well-separated Polaron defaults bipolarons hand (bipolaron )

.1 -rr

tt1ttt _

HOMO: bonding states (6-d) PPP and evolutio n of energy levels with p-doping (top) and n-doping (bouorn)

Figure IV-6. Representations of: (a) stable benzoic form; (b) unstable quinoid form; and (c) stable once-ionised quinoid form ofPPP. Note that to limit Figure size, the distance between defaults has been drawn as 3 benzene rings, instead of 5. Figure (6-d) represents the energy levels corresponding to structures of the: (a) fundamental state; (b) state with two neutral defaults; (c) once-ionised state and then finally to states corresponding to different levels of doping giving rise to simultaneous ionisation of more than one state, with n-doping for negative ionisation (bottom) and p-doping for positive ionisation (top).

use PPP as an example. In the neutral state, the stable structure of PPP is based on benzylic rings, as shown in (a) of Figures IV-6 and IV-7, and as corollary, a structure based on quinoids is unstable, as shown in (b) of Figures IV-6 and IV-7. As shown in going from Figure IV-6-c to Figure IV-7-c, when a default becomes charged, for example due to charge transfer, a quasi-particle polaron (c) appears.

IV Electron and band structures of 'real' organic solids

87

Energy

.-_

-_ -_ .. [ • . • • . . .I~.~i~~~.~~~.~~~~.~~I~.~~~~)!~.~~~.l •••• • • j

.~" \

Ionised and stable quinoid form. as F..... - E,... > E... (gain in E...)

I- -

- - t -- \.-

, .-

\ \

(d )' ... .,. ~

Quinoid form unslable in non·ionised slate

Benzoic form stable at fundamerual state r:

Lattice distortion

Figure IV-7.Configuration curvesfor PPPchains in neutral and ionisedstates.

In Figure IV-7 which shows system energy with respect to geometric configuration, we can see that a chain based on quinoids is stable in its ionised state (curve c). In effect, the energy Edist due to chain distortion in going from benzoic to quinoid forms is more than compensated for by the relaxation energy Erel which results in going from an unstable ionised benzoic based state (curve d) to the same state based on quinoids (curve c). Polaron quasi-articles correspond to charge-lattice interactions which self-trap within lattice deformations that they cause. Calculations [Bre 82] have shown that these types of quasi-particles are localised over approximately 5 benzene rings. Figure IV-6 shows a localisation over three rings only, but this is just to limit the actual size of the diagrams-and changes nothing with respect to our understanding of the mechanisms involved. In Figure IV-6-d we can see the evolution of energy levels with increasing density of introduced charges (doping density) and of the Fermi level, which sits midway between the lowest unoccupied and the highest occupied levels .We can simply note for the moment that levels associated with two neutral defaults (shown in Figure IV-6-b) can be interpreted as coming from a bonding or anti-bonding combination of a nonbonding pair of soliton states stituated in the middle of the gap. However, we will look to the actual nature of these sorts of states in Section 7.

6 How do we cross from polaron-exciton to polaron? A simple way of envisaging double electrical charge injection is to think of a positive polaron and a negative polaron which appear opposing electrodes, as shown in Figure IV-8-a and b. With an applied potential field, each polaron migrates across the material towards each other until they eventually meet, at which point they form an excited but neutral species . (The electron and the hole are excited outside of the

88

Optoelectronics of molecules and polymers (a)

IL

10 hand

• I

HOMO band

Polaron P'

Figure IV-8. Electron-hole injecting givingrise to a polaron-exciton.

HOMO and LUMO bands.) It is this species which is called a polaron-exciton and shown as a polaron-exciton singlet in Figure IV-8-c. In later Chapters detailing optical properties, we shall look more fully at the properties of excitons. Using PPP as the example, Figure IV-9 represents the generation of an intra-chain polaron-exciton (remember also this occurs over 5 rings, but 3 are used here to fit in the representation). We can see that there is a strong coupling between electronic and vibrational excitations as each inserted charge results in a strong geometric distortion of the lattice, which displays a quinoid based structure. When a radiant combination occurs, sometimes photons with an energy higher than that which separates energy levels of two polarons (or bipolarons) are formed, indicating that lattice coupling and therefore any induced distortions are less than that envisaged by previous models of polarons or bipolarons. The charges which make up the exciton, which is no longer a polaron-exciton (!), are thus tied to each other by an energy of no more than several tenths of an electron volt. This can be due to several factors (which will break any symmetry to the system Hamiltonians) which are subject to numerous controversies [Ore 95, page 46].

7 Degenerate n-conjugated polymers and solitons [Hee 88] In defining the soliton, which appears within chains of degenerated polymers such as polyacetylene -(CHh-, we will have defined the complete range of quasi-particles. As shown in Figure IV-lO-a, polyacetylene is obtained in the cis form directly after its synthesis, however, with increasing temperature it tends towards the trans form . During the isomerisation process, with increasing temperature, faults appear in alternating double and single bonds . These bonds occur during 'dimerisation' as we

Exciton

Figure IV-9. Representation of intra-chain excitonin PPP.

IV Electron and band structures of 'rea l' organic solids

89

(b)

Thermodynamic ally stable trans form

Cis form (a)

• signifies soliton (q = 0, s = Y2) at defa ult in double and simple bond alternation

Figure IY-IO. (a) Cis - tran stransforrnation in -(CHh -; (b) soliton type defau lt in -(CHh chain.

have seen in Chapter III, Section II-I . Defaults can arise due to the chain being energetically equal on either side, as if two configurations could equally be formed , as is shown in Figure IV-I O-b. It has been estimated that such types of defaults can typicall y form once for every two thou sand CH units. The state is, from an energetic point of view, equivalent to a dangling bond , with a 2pz configuration for the lone electron which makes the soliton shown in Figure IV-II-a. And is therefore midway between rt- and n "-levels, as shown in the left hand side of Figure IV-II-b. As shown on the right hand side of Figure IV-Il-b, the Peierl s distortion form s a band gap between full and empt y bands. Figure IV-II-c shows the state density function where we can see that the soliton (or dangling bond) is situated in the middle of the band gap. The Figure can be compared with that of Figure Il-5 for a dangling bond in a tetrahedral carbon system. Overall, the chain remains electrically neutral, as the carbon atom near where the soliton resides is surrounded by 4 electrons. However, the system can give a paramagnetic signal due to the unmatched electron having 1

s = 2'

I

.... 0 *

"r----;:-1·,

>~L r ~~ 1 I ··-;1

tM

C : 2s' 2p'

2sp'

I I I I

\..

++

It

"'j t+ t+ t+

~.J!~ ~~:> tt !

fa

I I I

lr;---:.l ---

L;;:le~~::~i~·· · · i ~·· · · ·t+·· · · · · · ~ ·~·~·~·~·~·· ·++·· ++·· equivalent to that of dangling bond

"---- -~ M,

M, (b)

C : 2s' 2p'

(a)

- - - LUMO

==== ".~~

a*

.

~~t;~~~~~~~~~?I~l~~.~

It *

.

dangling bond level

HOMO band

a (e)

N{E)

Figure IY-H. (a) Representation of bond formation between two groups -CH- ; (b) energy level evolution with increasi ng interacting monomers, going from 2 to 4 then to n; for a chain of n monomers Peierls distortion gives rise to gap EG; (c) resulting band scheme generated by Peierls distortion also showing level of dangling bond.

90

Optoelectronics of molecules and polymers

"~

~

2 neutral solitons

pclarons

+ - c:::::::::=:.- __

~leV

~~ eutral soliton

(dl

IW O

(e)

(b )

(aj

Charged soli Ion p

(e)

.t.

Two charged soliton, >0

Polaron P +

(f)

Rand open in g

c used by ioerea-ed degeneration

-+ =f

~rTwo polaron,

Two charged solitons < 0

Figure IV·12. Energy statesfavoured duringdifferent stepsindopingof - (CHh -, as followed in text.

Now we shall tum our attention to doping effects, with the following dopants: n-type donor (such as alkali metals) which can transfer an electron to -(CHh -; and p-type acceptor such as b or AsFs which are capable of taking an electron from a chain of -(CHh- . We shall take as example a single chain of -(CHh- on which there are two solitons corresponding to two excited states, as detailed in Figure lV-12-a. On doping, one of the two solitons is ionised and we thus obtain the associated pair charged soliton-neutral anti-soliton, as shown in Figure IV-12-b. Following their interaction we obtain two levels-one bonding and the other anti-bonding-which correspond to a polaron, and as detailed in Figure IV-12-c is of p-type with an acceptor-type dopant. If we increase the doping, two polarons can be formed as shown in Figure IV-I 2-d, which give rise to two charged solitons (Figure IV-12-e), and as the doping increases even more, a soliton band will appear and grow until the forbidden band is finally closed by around 30 % w/w of dopant (Figure IV-12-f) [Bre 82].

III Towards a complete band scheme including structural defaults, disorder and rearrangement effects 1 Which effects can intervene? As we have just seen, in real organic solids we should account for local interactions due to electronic charges and their associated lattice deformations and vibrations.

IV Electron and band structures of 'real' organic solids

91

In addition, such quasi-particl es (polaro ns, bipolaron s and solitons) are characteristic of the sort of dopin g used, and in contrast to classic cova lent semi-co nductors based on for example Si or Ge, doping atoms take up interstitial positions rather than substitute atoms in a latt ice. Th e specific electronega tivity of dopant atoms, generally alkali metals or halogens, is therefore used to transfer charge to materials such as rt-co njuga ted polymers. Studies of conduction phenomena have show n that in reality tran sport properties can be' isotropic and therefore charge transport and vibrational energies ca n only be considere d as one-dimensional over short ' local' distances. Hopp ing mechanisms for inter-chain movement of quasi-particl es have been introduced in attempts to account for real effects, such as Kivelson , Bredas or Zuppiroli models for solitons or networks of polarons and bip olaron s. Inter-chain tran sfer integrals have also been introduced. The se effects will be discussed when lookin g at tran sport mechanisms. While short range order ca n be discerned in crystals or certai n fibrou s polymers, in most mater ials there is a high degree of long rang e disorder which generally domi nants their properties. We can therefore con sider such materials to be amorphous, but recognise that there will be localised states appearing in band tails due to the random distributi on of potential well s from electron sites (as shown in the Anderson model) . Exa mples of these can be localised states in between polymer chains or associated with particularly amorphous zones of molecular solids.

2 Complete band scheme accumulating different possible effects a Construction of density state function We shall introduce vario us contributions in order to construct a com plete band scheme . Except in one-dimensio nal media, disorder and electron-lattice interactions do not strictly result in a simple sum of two effects, rather eac h acts in sy nergy to produce localised states . Energe tically speaking, this can only occ ur within the energy band gap (polarons) or in the mobilit y gap (disorder in the case of amorp hous semiconductors) (Emin in [Sko 86]). In order to obtain as genera l a band scheme as possible (Fig ure IV- l3 ), we must also associate with the localised states the state density function (Pfliiger in [Sko 86] and A Moliton in [Sko 98]) such that: • •

the introduced band tails increase with disorder (zon es C5 and C6 in Figure IV-13); and contributions from ele ctron-l attice interactions increase in polaroni c or bipol aronic band s with incre asing interactions and their electronic effec ts (zo nes C3 and C4 in Figure IV-13).

On considering the mechanism of charge transport , we can see that charge carriers can be thermally activated towards localised (between which there ca n be phonon assis ted hops) or delocalised (zo nes C I and C2) states. The energy involved increases in going from the former to the latter phenomen a, and eac h process appears with an

92

Optoelectronics of molecules and polymers E

Conduction band. delocalised stales

Central gap d fault hand

Valence band Delocaliscd states

tate density in default bands

tate density in electronic and polaronic bands

Figure IV-13. Band scheme with all envisaged energy levels. To the left are shown state densities resulting from localised defaults i.e. bond ruptures.

increase in temperature. For an intrin sicall y 'p erfect' material the energ y is equal ,

a priori, to the half-gap between valence (rr) and conduction (n") band s. We should note that there can be various perturbations due to thermal effects durin g synthesis, UV irradiation, ion impl antation etc. and structural default s such as vacancies, dangling bond s or chain ends, which result in conduction mechan isms at the band gap centre. Localised levels are thus introduced (zone C7) of a nature similar to those for carbon in Figure II-5. Geometrical fluctuati ons, for example changes in bond angles, increase the size of these band s, which in reality split into two band s (Hubbard ' s bands, zone C7).

IV Electron and band structuresof 'real' organic solids

93

b n-Conjugated polymers As elsewhere detailed for PPP [Mol 98], we can have in reality two types of default band s. Here we shall detail the energy levels formed on combining two carbon atoms which have hybridised states 2sp2 2pz as shown in Figure IV-I I-a. There are a-bonding and o"-antibonding orbitals in addition to Jt-bonding and rr" antibonding orbitals. If we form an electroactive polymer (M n ) from monomer (M) containing two carbon atoms , then as the polymer is constructed from alternating single and double bonds the n- and Jt*orbitals give rise to HOMO and LUMO band s (Figure IV-II-b) and the resulting state density function represented in Figure IV-I I-c. When a default associated with removal of an electron occurs, for example from a dangling bond following chain rupture, localised states which appear depend on the original orbital ofthe electron, which could be either a n-orbital (pz) or a a-orbital (sp2 state) situated (Figures IV-13 and 14), respectively: (i) at the centre of the gap (denoted Ep, == E p) separating HOMO and LUMO, and are the only states that we have considered up to now; or (ii) at lower levels than those mentioned in (i) and distributed in the neighbourhood of the primary sp2 level. Given that (E p - Es ) ::::: 8.8 eV and that Esp2 = ~ (E, + 2E p ) , we have:

In PPP, the highest point of the valence band is some 1.5 eV below the Ep level and the state which corresponds to a C - H bond rupture, Esp2 , is 1.4 eV below the same high point of the valence band, which allows charge tran sfer to occur from the highest point of the n-band to localised states . This charge transport process results in a decrease in the Fermi level, which otherwise would be at the centre of the gap, placing it nearer the valence band resulting in p-type conduction. Figure IV-14 details evolutions [Mol 94] of energy levels in a band scheme due to generation of dangling bonds (Figure IV-14-b) , which can equally induce hole s in the valence band (Figure IV-14-c) and can be considered as positive polarons with poor mobility due to their strong lattice attachment resulting from Coulombic interactions with negative charges on the polymer chain, as opposed to chemical doping. Figure IV-13, which represents the state density function contains not only a default band at the gap centre (zone C7), but due to the inevitability of CH ruptures in polymers, it also has a default band situated at lower energy values (zone Cg situated around level E sp2).

3 Alq3 and molecular crystals So far in this Chapter we have concentrated on the real behaviour of rt-conjugated polymers. Alq3, however, is not much different in its behaviour, as it is now widely thought that polarons do occur in molecular crystals in general (Karl in [Far 01]) as well as specifically in Alq3 (Burrows in [Miy 97]).

94

Optoelectronics of molecules and polymers

a e-antibonding

(a )

tc)

(b )

Energ) level»

• ub-eq nt r -arrang mem after a ·hond rupture

a ociated with default

generuuon

n -anti bonding

(pri or III rupture or o -

(lJ-l>ond rupture)

bond )

E, (imrinsi )

-oJ I

EP., ~""'

\

2p'

,

\

\

...

I I I

,

'.... -- ---- ~ '-¢=~).,L~ --'- ----t·---· - .----.-.---.-.- - \

5 ~~--

1.~... . .~ . ,.~ "'~:

-' , ::J:

I

I,

.."~ I

I

r--~--- --

____

I

': .

~: I

\ I

;~

"

~

lIalf·fuli

-, ~

lrnportant 10 note ~ that . p' level i, E. well below I vel p.

T

(

II

/I

localised dangling a -bond

abonding

i,X, JeSCt ,

,

/I

H

1--

-

----11...-_ ..Il.......L.._ _--.

G"ncr:lled po ii i ve polaron ha, poor

mobility due 10 Cuulombic interaction with nc,al i\'C charge on chain.

Figure IV-14. Energy levels in a n-conjugated polymer which can appear following rupture of C-H or C-C bonds, due for example to thermal effect, irradiation or ion implantation.

a Structural defects due to a charge presence in Alq3 Structural defects resulting from electron injection into Alq3 which becomes an anion have been studied. According to Chapter III, Section III-2 , an injected electron should enter the first empty level locali sed around pyridine groups. Calculations have shown that Al-O bond lengths should remain unchanged while Al-N bond lengths should change con siderably, as the presence of an electron on a quinolate ligand on an

IV Electron and band structures of 'real' organic solids

95

anionic Alq3 induces an increased interaction with the central cation. The AI-N bond between the host ligand and the central cation should shorten. The host quinolate ligand will though in tum generate repulsions between it and other quinolate ligands due to extra negative charge, and two remaining AI-N ligand bond lengths will accordingly increase . In order to approximately define the trapping energy associated with an electron injected into Alq3, calculations were performed using a neutral Alq3 of the same geometric structure as its anion. Transition energies 3.07 to 3.20 and 3.34eV underwent a high red shift of the order of 0.21 eV with respect to the neutral and therefore fundamental state of Alq3, representing the electron trap energy depth . It was also thought by Burrows , in the following manner, that these calculations permitted a further precision of the nature of the traps in Alq3. When an electron is localised on a molecule it induces anti-bonding orbital population and the molecule accordingly relaxes into a new structure (Frank-Condon principle) . This relaxation towards a lower energy level can be used to automatically trap an electron which could move to a neighbouring molecule through polaronic processes (see Chapter V). Given that there are many possible variants to the Alq3 structure and accordingly possible calculable energies , due to isomers and thermal vibrations, an exponential breakdown of trap levels leading to distributed levels in the LUMO band is possible. Conduction mechanisms associated with trap levels can therefore appear in otherwise resistive media .

b Molecular crystals With localisation of a charge carrier, there is a polarisation of its host molecule and its close environment, and due to the influence of a weak (see page 267 of [Pop 82]) polarisation field, the carrier is assimilated with a polaron . When moving, the polaron must overcome a potential barrier associated with potential wells it itself induced in the lattice. Introducing the notion of trap levels, which correspond to distributed localised levels following usual functions (in particular Gaussian), appears necessary once molecular crystals undergo a transition to an amorphous phase, as discussed elsewhere (page 245 in [Pop 82]), and for example observed in vapour deposited thin films of tetracene at T < 160 K.

IV Conclusion Taking into account all possible electronic levels that appear in an organic solid can at first seem complicated. In order to simplify the complexities presented in Figure lIll l , Figure IV-15 sets out practically all envisaged levels for Jt-conjugated polymers. As for amorphous semi-conductors, we can see that there are foot bands (for n- and Jt*-bands) and that states can be just as well associated with any type of disorder, as can charge carriers with the lattice (i .e. polarons).

96

Optoelectronics of molecules and polymers

It * -conduction band

Fenlli energy for n-doping : polaronic and/or stares a. ociated with disorder

EA Gap centre localised Slates

1;, Fermi energy for p-doping: polaronic and/or tate a.. ociated with di Ncr

It-\ ale nee

hand

State density

Figure IV-IS. Simplified band scheme and positionof EF following doping of n-conjugated polymers.

We shall try and determine the mobility that charge carriers can have in the 'foot' or lowest band states . In any case, it must be relatively poor and some of the states tend to behave as though they were like traps in a manner analogous to localised tail band states in amorphous semiconductors. Trap levels and their behaviour will certainly be studied in molecular solids such as Alq3 and as we have seen here, the existence of the trapping levels is equally associated with a lattice relaxation associated with the charge presence. Even if it is not possible to consider an universal band scheme for all real organic solids, we can at least now conceive that carrier localisation, on obligatorily distributed energy levels to account for the disorder that each carrier brings, is a common characteristic of these solids. The nature of and treatments to which these materials may be subject (thermal, irradiation) result in a diversity of localised states. Finally, we can state that transport properties of real organic solids, as in all materials, are conditioned by: •

the position of the Fermi level in the band scheme, as it conditions, along with other effects, the type of transport mechanism which mayor may not be thermally active ;

IV Electron and band structures of 'real' organic solids

• •

97

the value of state densities as a function of energy-which determines the number of charge carriers; and the nature of these states-on which depends the value and expression of mobility.

We shall now go on to try and define more exactly the laws which characterise the various possible transport mechanisms in Chapter V.

v Electronic transport properties: I Conduction in delocalised, localised and polaronic states

I Introduction In this Chapter we shall establi sh the different laws required for electrical conductivity within a band scheme for organic solid s (Appendix A-6 summarises laws for thermoelectrical power) . Organic solids can be deemed semiconductors, or in more general terms conductors, especially when considering certain rr-conjugated polymers undergoing development at the present time. The laws depend upon the nature of the conduction states involved, which in turn may be due to: • • •

•

valence and conduction band delocalised states (HOMO and LUMO bands in strong bonds) ; localised states associated with disorder that governs the depths of wells ; states localised around the Fermi level in which transport mechanisms by hopping (Hopping to Nearest Neighbour, or HNN) or tunnelling effects (Variable Range Hopping, or VRH) appear; and polaronic states due to the localisation of carriers in wells with depths dependent on the ability of the lattice to deform , and with different laws appearing as a function of temperature domain .

The nature of the states implicated in conduction depend s on the position of the Fermi level, which is typically situated half-way between the highest occupied level and the lowest unoccupied level. We shall thus describe, with respect to the Fermi level, a scale of energies within a general band scheme in order to characterise the theoretical form of conductivity in each of the states listed above . We shall not go into conduction mechanisms specific to organic solids and insulating states , in which conduction can be limited to space charges and /or intermediate trap levels more normally associated with OLEOs , but rather leave this and conduction limited to electrodes and interfaces to later Chapters.

100

Optoelectronics of molecules and polymers

II General theories of conduction in delocalised states: Boltzmann's transport equation and the Kubo-Greenwood formula for mobility (see also Appendix A-4) 1 General results of conductivity in a real crystal [Que 88 and Sap 90]: limits of classical theories The classical theory of conductivity (a) was initially elaborated by Drude using a 'billiard ball' model. The relaxation time (1:) (definedas the average interval between two successive collisions undergone by an electron) was assumed to be the same for all electrons, and was used in the following eqn (l): nq2

a

= -1: = ao m*

(1)

in which n is the concentration of electrons with effective mass m", In an alternating field with period (eo), conductivity is given by the relationship nq21:1 - iW1: + w2 t 2

a(w) = m*l

(2)

The real part is thus written: aR(w)

nq21:

= --

m» I

I

+

W21:2

= ao

I I

+ W21:2

(2')

and is valid at high frequencies although tends towards ao at low frequencies (W1:

«

I).

For a metal, the problem can be resolved in reciprocal space using Fermi-Dirac statistics and globally gives the same result [Que 88]. If we go on to suppose though that the relaxation time is the same for all electrons, we find that only electrons which have an energy of the order of the Fermi level participate in conduction, remembering of course that the concentrationof the n electronsintervenesin determinationof a. The relaxation time 1: shouldindeed be denoted as 1:(EF) as it is related to electrons in levels neighbouringEFin the case of metals, degeneratesemiconductors and semiconductors for which EF is in the conduction band. The hypothesis that 1: is the same for all electrons, however, is subject to dispute as the relaxation time of any diffused electron depends upon its velocity v. If at is the section of total efficientdiffusion, for isotropic collisions, and N is the number of particles struck per unit volume, the probability P of a collision per unit time for an electron at velocityV is such that [Smi 61] P = ~ = Nvo., in which 1: is the relaxation time such that L = V1:, where L is the mean free pathway. Thus, in reality, t depends on v and thus k. Equally, 1: also depend~ on the direction of electron displacement, _ and generally can be written in terms of k. If an electron is diffused by a particle, and the wavevector k varies by a value ik « k, then the hypothesis that t = constant is acceptable, bowever, in the alternate scenario we must take into account the dependence of 1: on k. In order to perform the required calculations, we can use the state density function (N(E)) within energy space.

V Conduction in delocalised, localised and polaronic states

101

2 Electrical conduction in terms of mobilities and the Kubo-Greenwood relationship: reasoning in reciprocal space and energy space for delocalised states (cJ. Appendix A-4) a Hypotheses used for calculation The actual calculations used, detailed in Appendix A-4, were initially developed for inorganic semiconductors. Boltzmann's transport equation is, however, a semiclassical equation, due to its use of Newton 's law which demands that particles have precisely known positions and moments . In turn we are required to reason in terms of phases in space in which the probability of the presence of a particle is denoted by the function fo(i~, r, t) . In addition , quantum mechanics is broug~t to bear by considerin~ the levels of collisions which are supposed to vary the vector k instantaneously by ~k without a variation in the position of charge carriers . From a practical point of view, this results in assuming that the collision duration is negligible with respect to the interval between two collisions for any given particle of energy E, which corresponds to a value of k given between two collisions. We should also note that the aforementioned hypotheses of relaxation time is formulated such that following a perturbation the system will revert to its equilibrium, following an exponential law characterised by a relaxation time teE) . When isotropic collisions occur, it has been shown that the relaxation time is equal to average times between collisions [Lun 00].

b Formulae for mobility If the applied field Ex, with respect to Ox, is uniform and constant, we obtain for current density (Ix), again with respect to Ox, the following expressions: •

in k space (from eqn (13) of Appendix A-4):

(3)

•

in energy space (from eqn (14) of Appendix A-4):

J, = q 2 Ex

i

E

For isotropic diffusion, in which

t(E)v~

--foO - fo)N(E)dE. kT

(v~

= 4)

(3')

and where parabolic energy bands

exhibit minimum k = 0, we can use an approximation of effective mass (m") which is such that

102

Optoelectronics of molecules and polymers

(see Appendix A-4, Section III): -

-

when L(E) = t is constant, we obtain Jx = nqj.LE x , in which j.L is the mobility of charge carrier q [eqns (16 and 22) in Appendix 4] given by qt j.L = - ; (4) m* when the system is degenerate (Ep in the conduction band), we obtain to the first order [eqn (23) in Appendix 4] j.L ~ qt(Ep) ; and m*

-

(5)

when teE) corresponds to non-degenerate charge carriers (distributed according to Boltzmann's law) nq2

(t} , or j.L = -

= -

(J

m*

Jo

OO

q

m*

(r) , with

t(E)E 3/ 2 exp (-~) dE

Jo

oo E3/ 2 exp (_ k\) dE

(t) =

=

(Et (E)) (E)

(6)

This average value for teE) is sometimes denoted ((t (E))) to clearly indicate that it is an average obtained not only from the single component of the distribution but also from EN(E), which can function N(E) = N(E)f (E) ~ E 1/2 exp (only appear in calculations specific to transport equations.

k\),

c The Kubo-Greenwood relationship a-General case

Eqn (3') permits writing the conductivity in the form: (J

= q2

i

E

t(E)v~ --fo(l - fo)N(E)dE, kT

(7)

which in tum can be written as the Kubo-Greenwood formula: (J

= q

j.L(E) = ~ -Case

(v~

f

j.L(E)N(E)f(E)[l - f(E)]dE , with

qt(E)v~

kT

(7')

(8)

of isotropic diffusion and the effective mass approximation

= 4= 3~' ) :

on introducing this expression for v~ and noting that E = ~ kT in eqn (3'), we again obtain the Kubo-Greenwood equation (7') in which

qt E j.L(E) = --=-. m*E

(9)

Eqn (9), without approximation over fo, is more general than eqn (5) which is applicable only to degenerate semiconductors.

V Conduction in delocalised, localised and polaronic states

103

III Conduction in delocalised band states: degenerate and non-degenerate organic solids 1 Degenerate systems For degenerate systems we must suppose that EF enters the conduction band and that we are dealing with ' metallic' conductivity. Figure V-I represents the conduction band in a disordered organic solid with delocalised tail bands . In this degenerate system we can imagine, for example, that the high concentration of charge carriers, which allow the Fermi level to enter the conduction band , is due to a high degree of doping of the n-conjugated polymer which is 'highly charged ' . In addition, in order to imagine a conduction which can occur within a band of delocalised states , we should be using a regime of weak locali sation, in other word s, the mean free pathway L ::: a, and Vo/B ~ I (ef Chapter II, Section IV-3).

a Example of a degenerate medium practically without disorder (£

»

a)

Conductivity can be expre ssed using Drude-Boltzmann's general formula,

(JB

=

qnu = nq2~SEF) , in which n is the concentration of electrons and It is given to the first order by eqn (5). t(EF) Repre sents the relaxation time relative to the Fermi level. With I = vFt(EF) in which VF is the velocity of charge carriers (at the Fermi level) and I is the mean free pathway we have (JB = nq 21 / m*VF. On introducing the crystalline moment, ft kF = m*vF, we obtain (JB = nq 21 /ft kF, in which kF is the wavevector at the Fermi surface. If we now consider the system at absolute zero , we can quickly obtain n as a function of kF using reciprocal space . With the number of cells , of volume 8n 3 /V = 8n 3 for V = I, at the interior of the Fermi sphere being ([4/3]nk~) /8n3, and in taking electron spin into account by introducing n = 2([4 /3]nk~) /8n 3 for doubly occupied cells, we obtain metallic conductivity written in the form (JB = 4nk~q21 /l2n3ft . If we set SF = 4nkl at the Ferm i surface , we obtain the following widely used and general formula [Mot 79]: (10)

E

Figure V-I. Position of EF in degenerate and disordered organic solid for conduction in delocalised states showing conduction band and tails bands of localised states.

104 Optoelectronics of molecules and polymers

b Lowleveldisorder andthe effectof weaklocalisations (£

~

a)

a The central principle Consider first of all the effect of weak localisation of the conductivity detailed just above . If E decreases following collisions, multiple diffusions occur. If in addition, the collisions are elastic , generated for example by dopants or impurities spread throughout the material in a random fashion, the wave funct ion (\II) does not loose its phase memory in a regime of weak delocalisation (£ :::: a). Additionally, if E is sufficiently large with respect to the wavelength of the wave functions (£ » ).), constructive quantum interference can develop between waves which are following different paths, as shown in Figure V-2. Under these conditions, a backscattering appears along with a diminution in conductivity, which follows the form: a = aB { 1 -

(kF~)2 } .

(10')

To generate the backscattering, E must be sufficiently large (£ > )., or Lf). > I, and with k = 2rr/)., k£ > 1) to create interferences (& = m), with integer m), and yet must also be sufficiently small so that colli sions occur to a non-negligible degree, without which £ :s a (and k£ :s 1) and we would have a localisation strong enough to provoke a loss in memory of the wavefunction. To sum, this compromise can be reached under a system of weak localisation with E :::: a. The quantum interferences are well detailed in [Ger 97], for example. ~ Limiting phenomena of quantum interference Quantum interferences can only occur if the interval between 2 elastic colli sions t is much less that that between two inelastic collisions (which modify k and therefore also the wavefunction phase). The reduction in conductivity decreases when the number of interfering quantum sequences decreases (phonon effect at increased temperatures). The conductivity can then be given by a = aB{1 - C(I -1 /L) /(krl)2} [Mot 79].

site or constructiveinterferences

Figure V-2. With elastic collisions, two waves corresponding to two directions in a pathway return to 0 in a coherent fashion, resulting in interference, localisation at 0 and decreasing conductivity.

V Conduction in delocalised, localised and polaronic states

105

The term (1 -IlL) represents the reduction in diffusion when all collisions are not elastic and L can represent the length of the sample being studied, the length of inelastic diffusion, or even the cyclotronic radius when a magnetic field (H) is applied. In effect, a magnetic field induces a dephasing of trajectories and therefore implicates wavefunctions associated with diffusion transitions; constructive interferences can be made redundant and weak localisation effects diminished; and conductivity increases resulting in positive 'magneto-conductance', or rather, negative 'magneto-resistance'. We can now consider effects due to variations in potential and reinforcement of disorder. When Vo > B or, at the limit, Vo "-' B, the state density function exhibits an increase in energy and also a reduction in the state densities denoted by g for a value corresponding to the band middle (disorder results in state density functions N(E) being flattened to gN (E». In the principle part of the expression for conductivity-which is proportional to the square of the state density function, as detailed in Eqn 2.18" in [Mol 91]-where a correcting term intervenes, q2 is replaced by (gq)2 and thus the conductivity can be written (16) This last expression for a corresponds therefore to I ~ a and Vo ~ B, and relates the expected Anderson transition from metal to insulator. In addition, we can also note that as Vo increases, g decreases coherently with conductivity.

c Law for the variation in conductivity with temperature It has been shown in 3-D by Altshuler and Aronov, (detailed in Chapter 5 of [Mot 93]) that at low temperatures, where quantum interferences are important, the expression for a above becomes

Ke (kT) liD 2

a = aR(To)

+

4rt 21i

1/2

= GB(To)

+ mT 1/ 2,

in which To is a reference temperature and K .a constant [Ell 90]. This law is established using L = V'tj and 'tj <X T-P, with L equivalent to the indice i corresponding to inelastic collisions which when dominating phonons give rise to p = 2, 3 or 4 depending on the temperature. As long distance electron-electron interactions are introduced, a minimum in the state density function results in E = EF.

2 Non-degenerate systems: limits of applicability of the conduction theory in bands of deIocalised states for systems with large or narrow bands (mobility condition) a Representation and properties of non-degenerate systems In non-degenerate systems, the conductivity corresponds to that of a semiconductor, in which EF is sufficiently less than Ec so that an approximation of the Fermi-Dirac function can be made using Boltzmann's distribution (see Appendix A-4, Section III-2-c).

106

Optoelectronics of molecules and polymers

The Fermi level must therefore be situated midway in the gap between valence and conduction bands (Figure V-3). In analogy to the terminology introduced by Fritzsche [Fri 70] of an 'ideal amorphous semiconductor' we are considering an 'ideal'system in the sense that we assume that there are no defaults giving rise to numerous, deep states in the band gap. These defaults, localised and degenerated in the gap can arise from impurities and dangling bonds, and the conductivity in these states, often encountered in organic systems, will be detailed in Section 4. Initially we will suppose also that states in the band tails are of low density even if conduction through these delocalised states can occur. Conductivity can be obtained using the general relationship of Kubo and Greenwood: (J

= q! N(E)I-L(E)f(E)[l - f(E)]dE,

which gives eqn (6) for non-degenerate systems. In a more or less classical fashion, we can equally use the simplifying, following hypotheses: -

N(E) ::::::: constant in extended states, N(E) ::::::: N(Ed I-L is zero in the mobility gap Ell and equal to an average value (I-Le) in extended states (with Einstein's relationship, we can consider that I-Le

= ~C), and thus

following the classical law for semiconductors with

(J

= (JO exp(-

E, - EF kT ).

Comment J In semiconductors, classically speaking, the diffusion constant (De) is tied to the diffusion length L, by L~ = Dc'te. With V e = llt e and L, : : : : a, we have De = vea 2 . If diffusions are isotropic, i.e. equally probable in all 6 directions of a triangular pyramid, we have therefore in one direction De = (vea 2 ) /6. We can also

I

(E)

";': mubility I:a p

E

Figure V-3. An 'ideal' system can induce semiconductivity in delocalised states.

V Conduction in delocalised, localised and polaronic states

107

obtain , using the nearest neighbour approximation, as used in obtaining eqn (I5) below, vel' ~ n/ma 2 ~ 10 15 s-l . Comment 2 In amorphous semiconductors, Ec - EF = ~E - oT and

in which ~E

= EdT = OK) -

EF(T

= OK).

We shall now see that, giventheproperties oftt-conjugatedpolymersand organic solids, such as mobility values , band size in terms of the resonance integral ~ , how the possibility ofconductionthrough the delocalised band states, developed in II, is barelyrealisable [and yields eqn (6) for non-degenerate systems, i.e. semiconductors to which approximations for effective masses can be applied] .

b Condition on band size allowing application of effective mass approximation To define an effective mass, the size of permitted band s (8) must be B » kT, otherwise when the band attains a value close to kT (0.0026 eV at ambient temperature) it is not only the lowest band levels which are occupied but also all levels inside the band due to occupied by thermal effects . The form of the obtained mobility, for a strong 2

bond, is no longer acceptable i.e. that of eqn (II) of Chapter II, Il- = q~~ ~,obtained with the mobility relationship Il- = ~: with a simplified version of

t

in which was

introduced the effective mass rn" = £~2 Z from eqn (8) of Chapter II. So, to obtain an expression for rn", we need to use an approximation which involves only the lowest states in the band . Here, k ~ 0 and is centred in the zone for which E is at a minimum (i.e. [ ~~ ]k:::OO = 0), allowing the use of a cosines approximation for energy in a strong bond .

c Systems for which the permitted band B is large (B » kT); n-conjugated polymers From the preceding paragraph, we can see that the approximation of effective mass can only be applied to large bands (8 » kT) , It is here that the mobility of charge carriers within HOMO and LUMO bands in n-conjugated polymers can be evaluated using (II) In addition, we should note that as detailed in Chapter II, Section II1-3-8 , so that bands conserve a physical significance, B > ~E in which ~E ~ ~ [Chapter II,

108 Optoelectronics of molecules and polymers eqn (20)]. Under these conditions, to obtain conduction in these large delocalised bands, the following should be valid: qta 2 .0.E

qta 2 B

qa 2 I

(12) !i2z > !i2T ~ ---';Z· The inequality in eqn (12), It > q~ i,is the final condition for conductivity in It =

2

delocalised bands B. With a of the order of several A, which is about the length of a strong bond in a n-conjugated polymer, and Z ~ 2 we obtain the condition It > 10- 1 cm 2 V-I s-I . As the typically observed value for conductivity in these polymers is of the order It > 10- 4 cm 2 V-Is-I, we can conclude that that transport probably does not occur within these delocalised bands . We have to consider that in n-conjugated polymers any charge mobility due to delocalised states is well reduced by the passage of charge carriers through more localised states, such as intermolecular states. These, for example, include hopping between polymer chains with a pathway possibly tied to interchain polarons , such as ' intrachain or chain-end defaults' which can be due to impurities, faults in conjugation, dangling bonds and traps of all sorts. Mobilities associ ated with these sorts of transport mechanisms shall be detailed later.

d Systems with narrow permitted bands (B < kT): small molecules [Wri 95] Molecular solids, as we have seen in Chapter III, Section III, present narrow permitted bands , as intermolecular interactions operate through weak Van der Waals bonding and a limited overlapping of molecular orbitals . Given the argument presented in the preceding Paragraph b, we find that the effective mass approximation can no longer be applied to an attempt to evaluate charge mobility. In assuming that any collisions is isotropic and therefore relaxation time is equivalent to the time between two collisions, we can use the general equation for mobility detailed in eqn (8) (see Section II-2) . We therefore have £ = tv x . We can thus derive (following [Wri 95]) that It =

*

k~(tV~) = ~~ (vx) .

As v« = t~ [Appendix A-4, eqn (1)], relative to the size of a permitted band we have .0.E ~ B, and while .0.k ~ l/a (corresponding to the order of a permitted band of height B and Brillouin zone in reciprocal space, comparable to the simple representation in Chapter I with E = f(k)) , we obtain Vx terms q£ Ba It = kT h '

~ ~a, or in other (13)

At this level, we can in fact reason in two ways, both of which arrive at reasonable results : •

either that to have a conduction through delocalised states, we need E > a (from the second conduction given by Ioffe and Regel, detailed in Chapter III,

V Conduction in delocalised , localised and polaronic states

109

Section III-3-b ), resulting in the need to verify that: (14) And for small molecules with a band size B that for the polymers:

~

kT, we have the same condition as

(IS )

•

On taking a ~ 5 x 10- 8 em (where a repre sents intermolecular distances, which are slightly longer than covalent bonds and we can reasonably assume that a 2 is at least an order of differen ce greater for small molecules than for polymers), we should have u ~ I to IOcm 2 V- Is-I. With u ~ I cm2 V-I s- J and a ~ 6 x 10- 8 em for anthrac ene, we do not have the inequality requi red in eqn (IS) i.e. !-.L ~ 5 cm 2 V-I s" : or with the known mobility (!-.L ~ I cm2 V- 1 s- 1for anthracene) and B ~ ~,inwhich the resonance integral for anthracene ~ ~ 0.0 leV with a ~ 6 x 10- 8 em, we can estimate using eqn (13) that [ ~ 3 x 10- 8 em. The free mean pathway appears to be considerably less than the intermolecul ar distance a, itself incomp atible with conduction through delocalised states (Joffe and Regel).

IV Conduction in localised state bands Here we consider systems with many localised states and corre sponding to two particular cases: •

•

System I which classically corres ponds to a real (non-idea l), amorphous semiconductor with localised states esse ntially induced by disorder, thus with a low degree of occupati on, such that the Fermi level is below the band tails (Figure V-4) and conduction is in a band of non-degenerated localised states; and System 2 which has numerous defaults (such as impuriti es, dangling bonds and so on) which introdu ce localised levels deep within the band gap sufficiently occupied by carriers so that the Fermi level is found in this band (Figure V-5) and corresponds to a condu ction in a band of degenerated localised states.

_L...-

---. E

Figure V-4. Position of EF in conduction dom inated by band tail transport.

11 0

Optoelectronics of molecules and polymers E ~///

I'// / / / / ;'/// / // ;'///// / ;'/ / / / //

/ / / // / / / // / // / / / / / / /

h • . / // // / / / 1'; / // / / / // / / / / / //

N (E)

'-------------.

Figure V-5. Representation of conduction in a band of degenerated, localised states; regions marked with oblique and square lines correspond, respectively, to unoccupied and occupied states.

1 System 1: Non-degenerated regime; conductivity in the tail band In orga nic solids, and in particular in rr-conjugated polymers, disorder effects can result from geometrical fluctuations such as the rotation of a cyclic group or a bond about another bond yielding distortions in a polymer. The resulting fluctuations in potential are, in principal, quite weak (resulting in shallow wells) and operate over relatively long distances. The overall result, thro ughout the conjugation length of a polymer, is an increase in the distribution of nand n " states close to band limits. Weak localised states are thus generated at the band edge . The system correspo nds to that show n in Figure V-4 which is non-degenerate and fo(1 - fo) ~ exp( - [E - EFl/kT). With wavefunctions being localised, conduction can only occ ur through therm ally activated hops. In eac h pass from one site to the next, the carrier receives energy from a phonon. Condu ctivity is given by Kubo-Greenwood 's functions, as shown in Appendix 5. We still have (J = q J J.L (E)fo(l - fo)N (E)dE, but this time around, J.L = fLo exp (-

~)

and fLo =

~~~

(see eqn (7), Appendix 5) in which U represents the

energy barrier betwe en localised states and TO is such that T'o = Po = Vph in which Po represents the probability of a transition from one state to another without an energy barrier (i.e. if U were equal to zero). Taken into account is the frequency of phonon s ( Vph) which are quasi-particl es that stimulate transitions between localised states . t corresponds to the mean free pathway between two hops, and in some senses is equivalent to the mean hoppin g length for localised states conventionally denoted R. If we take the typical value for phonon frequen cy to be Vph ~ 1013 s- I and U ~ kT, we obtain, at ambient temperature, a mobil ity of the order of 10- 2 cm2 V-I s- I, a value lower by two orders than the mobil ity found in extended states. Typically, the analytical representation of the band tails is given by a state density function of the form N(E) = (E~~~~)S (E - E~ ) in which s = I (linear variation) or s = 2 (quadratic variation) . In making x =

Ek"i

A

,

and ~E = Ec - EA, we

V Conduction in delocalised, localised and polaronic states

III

obtain [Mol 91] :

t.E

in which C, = faIT XS exp(- x)d x can be calculated by integration of parts with oo the function r es + I) = fo e- xxsdx, which is such that I' (s + I) = sf'(s) = sl. We finally obtain :

(J

{EA- kTEF+ U }

kT ) S = (Jloe = (JOI ( 6.E C, exp -

(16)

with (JOI = qN(Ec)vph \~\T = q2N(Ec) \JphR2 (here using R as the classical representation for the mean hopping length ), a factor independent a priori from T,

2 System 2: degenerate regime; conductivity in deep localised states System 2 occ urs when a material contains a high concentration of defaults and presents a high density of localised states in the band gap. Electronic transport is effected in this band by hoppin g from an occupied to an empty state with a contribution from a phonon. A high degree of disorder can be introduced by numerous chemical defaults, such as impurities, dopants, (un)saturated dangling bonds and rearrangements following bond rupture (see Chapter IV). The resultin g highly localised variations in potential give rise to deep states within the band gap, in which a band of localised states is thus generated about the Fermi level (Figure V-5). Polaron transport will be detailed in the following Section V, and so we will not go into detailing possible lattice relaxations and the formation of highly localised and poorly mobile polaron s. The conduction mechanism in such a band was studied in particular detail for compensated semiconductors, in which a closest neighbour transport mechanism was a dominant feature , and in amorph ous semiconductors for which Mott propo sed a mechanism based on variable distance hoppin g. In both cases the transport was effected through two localised levels A and B separated by an energy 6.E . Conduction thus corresponded (Figure V-6 ( I» to a thermall y assisted 'hopping' mechanism via a tunnelling effect (2). Except at high temperatures, this transport is favoured , as it only require s energy 6.E, with respect to hop (H) which , uniquely, is thermally assisted, and necessitates energy U » 6.E. In greater detail, the transfer of a carrier from site A to B needs three stages, as shown in Figure V-6: •

Transition I in which thermal activation of a carrier between two energeticall y equivalent levels with the help of phonon energy W = 6.E (the two sites A and B appear energetically equivalent to carriers which have this energy);

112

Optoelectronics of molecules and polymers

Be E· foE = fo'\ B

____ ~1C-J: --:~~ it

foAIl

E

I L.....:.-'l.-=-- -

i RAil i ~

:

RA~

B

Figure V-6. Scheme of transitions during conduction by hopping between localised levels at the order of EF. • •

Transition 2 for the movement of a carrier through a barrier, and corresponds to a tunnelling effect from one site to another of equivalent energy ; and an efficient electron-phonon coupling permitting the first step. When tlE/n is larger than the phonon pulsation maximum wmax , the coupling can only occur at the frequency Wmax ~ Vph ~ 10 12 to 10 13 s-l .

a Conduction due to Hopping to Nearest Neighbours (HNN) The probability (p) per unit time that a hop will occur between two neighbouring sites A and B is thus a product of the three terms listed just above. The Boltzmann factor, exp( -tlE/kT), indicates the probability of transition I due to the generation of a phonon with energy tlE. The factor required for transition 2, in which an electron transfers from one site to another, can be expressed as an overlapping factor in the form exp( -2aR) with wavefunctions being localised in \II = exp( -aR), in which R represents the spatial distance between adjacent sites and a the localisation length . The third can be taken into account with VphThus, from these three notations , we have: p = Vphexp (-2aR -

~;).

(17)

In addition , if N (EF) represents the density of states at the Fermi level, and does not vary to a practical degree within the limits kT about the level EF, then kT N(EF) represents the electronic concentration in the neighbourhood of Es (cf. AppendixA-5 , Section III). In taking eqn (6) from Appendix A-5, I.t = ~2 = pR 2k\ ' Kubo-Greenwood's relation ship gives:

r.r

(J

q = q 2 R 2 VphN(EF)exp ( -2aR - tlE) = qkTN(EF)pR 2 kT kT .

(18)

V Conduction in delocalised, localised and polaronic states

113

Comment

I.L can be evaluated by using Einstein's relationship, i.e. I.L = f~ ' in which D ~ p~2 However, this practise is difficult to uphold, as we are dealing with degenerated states , and the latter relationship was strictly established for non-degenerated states.

b Conductivity according to the Variable Range Hopping (VRH) model When kT attains a 'high enough' value with respect to t..E (typically within a high temperature range), a carrier has enough thermal energy to hop to an empty level of its nearest neighbour. This hop occurs with negligible spatial displacement-following on from the previous HNN model. However, if kT is of a low value with respect to t..E i.e. near low temperatures, the carrier can only hop to energetically close levels, even though they may be spatial speaking far away. A priori, the carrier therefore systematically looks for the energetically closest empty level even to the point of disregarding distance. The probability of transitions occurring due to tunnelling effects also diminishes due to an increasingly large barrier. We must therefore look for a compromise which optimises hopping distance giving by the condition

dP ] [ -dR R'optimised' -0 - .

(19)

The value of t..E must therefore be estimated as a function of R. In order to do this, we can consider that a hop over distance R, obligatorily, must occur within the volume (4/3) rrR 3, and that the number of energy states corresponding to such a hop have an energy of between E and E + dE given by dN = (4/3)rrR 3N(E)dE. If we consider only one hop between E and E + t..E (for t..E = t..AB, the length R = RAB) and dN = I = (4/3)rrR 3N(E)t..E (with E = EF), we reach ~E=

3 . 4rrR 3N(EF)

(20)

In addition, and in agreement with Mott [Mot 79] , if we take an average value for R, the mean hopping distance, then R

-

JJJW*rWdt

R = (r) =

8.'1'0

R

R 3

=

JJJW*Wdt

Jo

rR )0

r dt 2

r dt

3

=-R 4'

(21)

8.'1'0

the condition for optimised hops, 3 ] - 0 -dp ] ex -d exp [3aR - -- [ dR R'optimised" dR 2 4rrR 3kTN(EF) - . gives: R ' optimised' = R =

3 [ 2rraN(EF)kT

]1/4

(22)

114

Optoelectronics of molecules and polymers

Conductivity, in the form of eqn (18) can thus be written with the optimised value for R: ) 1/4 (23) (J = (JVRH = (JOvexp - T l/4 = (JOvexp - ; ,

( 8)

[ I

(T

with

a3 ] 1/4 ( 3 ) 1/4 , in which 8 0 = 2 = 1.66 Zn kN(EF) 3a 3 9 To = 24 [ ] , and (JOv = g2iP N(EF)\Jph = _g2R2N(EF)\Jph. 2n kN(EF) 16

8 = 8 0 - --

(24)

Comm ent

Eqn (23)comes about through reasoning in 3 dimensions (d = 3).The same reasoning with d = 2, or with d = 1, can give a general formula egn (23') in which:

with Y = [I / (d + I)] i.e. y = 1/4 in 3D, y = 1/3 in 20, y = 1/2 in 10.

c Use of Mott's formula (eqn (23») As R2 intervenes in (JOv,as in egn (22), and R2 = f (T- I/ 2), eqn (23) can be rewritten: (J

= (JVRH = (JOvrhT -1 /2 exp (-To) T I/4

(25)

From which we find that: (JOvrh = A\Jph

IN(EF) . 2 9 ( 3 ) 1/2 q - a- ' With A = g 16 2Jtk :::::: 1.5\ 1/2'

4 3n) To = 2 ( 2

[kN~~FJ :::::: 7.64 kN~~F) '

(26) (27)

These values can be empirically determined from log (JJT = f (T- 1/ 4), the slope 1/4

of which is equal, in absolute value, to ~~3 ' and the ordinate at the origin (T --+ 00) directly yields (JOvrh. We can thus derive values for a and N(EF): •

from eqn (27) we have N(EF) = 7.64 k~o' which used in eqn (26) gives

a = (JOvrh (kTo) 1/2 ; A\Jph 7.64 •

(28)

from eqn (28) in egn (27) we obtain N(EF) = (

(JOvrh A\Jph )

3

(kT _ 0 ) 1/2 7.64

(29)

Y Conduction in delocalised, localisedandpolaronic states In amorphous silicon (aSi), we thus obtain at ambient temperature a (lOw cm'), and N(EF) = 3.97 x 1027 states cm- 3 ev " . There are two pertinent comments that can be made at this stage: •

•

liS

= 0.871

the preexponential coefficient aOvrh is often abnormally high, similar to the value ofN (EF) which we have deduced here. This can result from the value estimated for Vph in terms of a single phonon transition. B is though pretty much well defined, and reasonably we can fix cC I :::::: 10 A and thus deduce N(EF). Accordingly we can go on to deduce R optimised (typically ca. 80 A) ; and it is not because the representation In aJT = f(T - 1/ 4 ) is linear that we have Mott's VRH law! We shall see later that percolation 'hopping' can result in the same law.

d A practical representation of Mott's law for PPP doped by ion implatation The curve ofln aJT = f(T- 1/4) is shown in Figure V-7 for PPP films, implanted with caesium ions at energies (E) of 30 and 250 keY and flux (D) varying from 2 x 1015 ions cm- 2 to 1016 ions em- 2 [Mor 97]. We can see that the linear law is well adhered to over the temperature range studied, which tends towards low temperatures. Values calculated for To vary between 3 x 105 and 4 x 106 K, with To tied to N(EF) and a (the inverse localisation length) by eqn (27). The preexponential factor aOvrh generally yields incoherent results [Ell 90]. So, as proposed below, we shall use as a starting point a reasonable value of I nm for 1/ a, given that interchain distances are around 0.5 nm, in order to determine values

- 0.50 - 1.00 - 1.50

S - 2.00 '"'[§ - 2.50 ~

O!

f:-x

IE=30 keY ; D = 10

16

- IE = 30 keY ;0 = 5x10

15

ions Icm 2

1

ions Icm 2

1

- 3.00

.'eo - 3.50

.2

i

- 4.00 -4 .50 - 5.00

IE = 30 keY ;0 = 5

X

1015 ions Icm 2

1

- 5.50 ......................-.......~~~..........~.........~...............1 21.00 22.00 23.00 24.00 25.00 26.lXl 27.lXl 28.00

Figure V-7. Representation of In O'~T = f(T - 1/ 4 ) characteristic of Mott's law for PPP films implanted with caesium ions, with implantation parameters alongside eachcurve.

116 Optoelectronics of molecules and polymers for N(Ef), which reach from between 2 x 10 19 to 3 x 1020 states eV- 1 cm" . The result lies is of an acceptable order for localised states associated with a band of deep defaults .

V Transport mechanisms with polarons 1 Displacements in small polaron bands and displacements by hopping We shall now look at the problem of transport phenomena associated with the displacement of polarons which have an energy spectrum detailed in Chapter IV, Section II-4. Figure V-8 below schematises the corresponding energy levels. In the scenario in which 1 =1= 0 (1 is the electronic transfer integral), surplus carriers, a priori, are able to move from one site to another within the solid via two possible processes: •

•

The first is called the 'diagonal' process and corresponds to a tunnelling effect of a small polaron between adjacent sites with unchanged values for the population of each phonon mode (during transfer of charge between adjacent sites). Nq = N~ for all values of q, in which N q and N~ are, respectively, numbers of the initial and final occupation states of the phonon and are associated with the q vibrational mode number. This process corresponds to a simple translation, with a modulus equal or unequal to the footprint of the lattice , the charge carrier and the associated lattice distortion, without changing the vibrational movement of atoms from their equilibrium position s. These processes occur together (tunnelling effect concomitant with the charge and the deformation) with a displacement within the band of the small polaron. The phenomenon is exactly equivalent to that in a non-deformational lattice, in which the occupation of any site by a carrier can be considered using proper states within a Bloch type system. Within the strong bond approximation, the proper values gives rise to a band of size 121 exp( -S) (for a cubic crystal). The size of the band for a small polaron is particularly narrow, more so if the lattice cannot be deformed (exp( -S) -+ 1), and the mobility of polarons associated with such a process would therefore be very low. The second is called the 'non-diagonal' process and concerns tunnelling effects assisted by a phonon (hopping) of a single carrier between adjacent sites. The levels occupied by phonons change during the carrier's transition from site to site, and here Nq =1= N~ for certain values of q.

Finally we should note that the mobility of a small polaron is the sum of two contributions: an associated displacement in the small polaron band and the hopping movement of the polaron itself. If we consider a small polaron moving within a perfectly ordered material, we should note that the description useful for free carriers (or here, a very mobile polaron) which undergoes random collisions with vibrating atoms is no longer acceptable when the variation in energy of such a diffusion is of the same order as the small polaron band size . In addition , the mobility within the small polaron band can only dominate

V Conduction in delocalised, localised and polaronic states

Be

EI

117

ron I ,d prior 10

lattlee d rormallon

Ep: polaron bondln!: n rJO"

Eo = Er - E.. ·DjaroO!J)"

prom' ""ocinted "jib pond"

mobile nolumm,

E."

Figure V-8. Energy spectrum associated with polarons.

at low temperatures, even in an ordered crystal. We should also reckon on the band being extremely narrow « 10- 4 to 10- 5 eV) and therefore the hopping regime can be considered only for disordered materials-which is generally the case for the materials we are studying . Energy fluctuations between localised sites are generally estimated to be considerably greater than band sizes. So, only polaron hopping assisted by phonons will be discussed hereon.

2 Characteristics of hopping by small polarons [Emi 86] Charge transport in many polymer s is generall y supposed to be based on hops assisted by phonon s. The charge carrier moves between two spatial locali sations accompanied with a change in atomic vibrational states (following the ' non-diagonal' process detailed above).

a Single and multi-phonon processes The electron-lattice couplin g force depend s heavily upon the spatial extent (d) of its state: if the state is confined about a single atomic site, the coupling force is extremely high. For highly extended states though, the electron-lattice interaction is much weaker, resulting from electroni c states essentially only interacting with atomic vibrations of wavelength t.. greater than its characteristic length d. An electronic state confined to a single atomic site will give rise to a high electron-lattice interaction force. In addition, if the electronic state extend s over a large monomer, its coupling with lattice vibrations will be very weak. Finally, the transport processes brought into play depend on two factors : •

•

one is the precise intensity of the electron-lattice coupl ing. If (Ep/ hwo) « I then the coupling is weak, and if (Ep/ hwo) » I the couplin g is strong. (In general terms we can consider that hwo represents the maximum phonon energy); and the other is the energy difference D.. between the two sites involved in the transport . (If D.. = 0, the two sites are degenerate).

According to the values of these two parameters, transport mechanisms for one or more phonon s appear.

118

Optoelectronics of molecules and polymers

ex - Mechanismfora single phonon when (Ep/nwo) « I, and when Li < nwo. Here we have weakcoupling with an extendedelectronic state (;::;; I 0- 7 em) (if the coupling tends towards zero, we are going towards a rigid lattice). The hopping density is thus dominated by a process which brings into play the minimum number of phonons, which in tum favours only those with the highest energy. At low temperatures (kT < nwo) hopping mechanisms are thermally assisted by phonons (with very low energies ;::;;10- 3 eV), absorbed by a minimum number in accordance with the principal of the conservationof energy. However, the hops occur over long distances as the states involvedare highly extended. At high temperatures (kT > nwo) the hopping processes loose their thermally active character and their number increases with temperature following T", ~ Multi-phonon mechanisms appearing when (Ep / nwo) » I, both with Li < nwo and Li > nwo. The dominant process brings into play phonons with relativelylow

energies. A high number of multi-phonon type emissions contribute to the hopping process. Clearly, a strong coupling generate a higher number of phonons than that whichnecessitatesa conservationof energy, however, the actualenergyof the phonons remains relatively low. There is a strong possibility though of transitions occurring with this type of process. Given that Li cannot be excessively high and that the temperature cannot be too low « 10 K), the calculation for the hopping probability is reduced to a calculation of probability in the degenerated state (Li = 0) which we can multiply by the term for thermal activation, exp(- Li/2 kT). We should note though that this is not the only term to rely on temperature; there are two others which can appear: •

•

one is that tied to hops between degenerated states and corresponds essentiallyto small polaron hops.At low temperatures(T < Te/2 where Te is Debye's temperature), the process is dominatedby the interactionof charge carriers with acoustic phonons, which give rise to a non-thermally activatedmechanism of the variable hopping range (VRH) type. At higher temperatures, the hopping process is independent of the type of phonon with which the carriers interact, and appears as if thermally activated. After detailing its origin in Section b below, Section 3 will show the relevant calculations; and the other is related to conduction by percolation which occurs when a rising temperature increases the number of involvedsites (which can also have a higher energy) and thus the number of conducting pathways. In appropriate cases, this mechanism can result in a conductivity which varies with exp[-(To /T) I/4], in a form which resembles that given by VRH.

b Temperature dependence of number of hops (strong coupling) Here we shall consider a hop, assisted by phonons, between two strongly localised and separated states. With a charge localised on one of the sites, the deformation of atoms associated with this charge will also result in a potential well around the charge (see Figure IV-5). And this is indeed the polaron effect. In addition, the charge

Y Conduction in delocalised , localised and polaronic states

119

cannot move without these atoms modifying their positions . Figure V-9, which comes from [Emi 86] illustrates the processes which a carrier follows from moving to an adjacent site with a suitable atomic displacement. In Figure V-9 (a), both atomic and electronic charge displacements occur through a tunnelling effect (diagonal process) . However, in the case of strong electron-lattice coupling, this tunnelling is relatively unlikely as superposition of (atomic) vibrational wavefunctions is poor (i.e. exp( -S) is small) . During their vibrational movement, atoms can find themselves in configurations where the superposition of their (atomic) vibrational wavefunctions is augmented (in process b I) . The transport process can thus occur in 3 stages : • • •

first, atoms go through an appropriate displacement (bd; second, atomic and charge displacements together give a tunnel effect ; and thirdly, the displaced atoms relax towards their configuration of lowest energy.

Even though the tunnelling effect in process (b) is easier than proces s (a), these different movements are more energetic, as energy is required to distort the atomic displacements. In reality, if we give enough energy to the system, a charge can move without needing the intervention of a (atomic) vibrational tunnelling effect as is the case in process (c). In state (c I) atoms are in a configuration which presents , momentarily, two degenerated potential wells (of the same depth) . The carrier can use this 'coincidence event ' to go to an adjacent site through a tunnelling effect (process C2) . Following this, the atomic displacements relax (C3).

.

. ·, ..

........ ... , "

.... ·...... .

... ..........

"

4

...

(a)

" ~

(b)

-(e)

·.. -

"

INITIAL

•

a

FINAL

Figure Y·9. Schematisation of displacement processes at different temperatures: (a) at low temperatures; (b) at intermediate temperatures ; and (c) at high temperatures.

120 Optoelectronics of molecules and polymers During the latter (Cl) transition, no (atomic) vibrational tunnelling effects are necessary, only the electronic charge passes uses a tunnelling effect; this type of displacement is called 'semi-classical' . At the lowest temperatures, process (a) dominates. When the temperature rises, process (b) starts to intervene with an certain inactivated contribution for a number of hops . At high enough temperatures though the semi-classical process (c) predominates; a number of hops are thermally activated and the energy of activation is the minimum energy which we must generate for coincidence configurations (at two atomic sites) to occur.

3 Precisions for the 'semi-classical' theory: transition probabilities a Coincidence event The principle on which for this semi-classical concept is based is the event coincidence which permits the passage of a charge using only its tunnelling effect. In the quantum regime , we still have to consider uncertainty with respect to time and therefore any associated intervals during which the passage occurs . Within classical limits, the event occurs instantaneously (and for us, when two atomic sites are identical that means that the two local electronic levels are identical as in -Axj(t) = -AXj+l (t)) . If we suppose a charge carrier occupying a site on a crystal has an associated electronic energy level which is a function of instantaneous positions of atoms of the crystal, we can see that as the atoms continuously change position with their vibrational energy, then the electronic energy of such a carrier will also change . Amongst the variations of distorted configurations resulting from vibrating atoms, there is a probability (Pj ) that a situation will occur when the electronic energy of one electron at one site is 'momentarily' equal to that on a neighbouring site; such a coincidental event is represented in Figure V-tO, following Emin 's proposition [Emi 86]. While in classical physics a coincidence event can be considered as an instantaneous event, in quantum mechanics we need to accord a finite time interval. If the time interval during which the coincidence occurs is long with respect to the time required by an electron to pass from one coincidence site to the next (of the order of ~t ~ (iii ~E) ~ (li ll)), then the electron can follow the 'lattice displacement' (or rather, displacement of vibrations) and hop with a probability P2 = I when the coincidence event occurs , in a situation characteristic of an adiabatic approximation. However, if the time required by the electron to hop is long relative to the duration of the coincidence event, then the electron cannot always follow the 'lattice displacement' and the hop probability reduces to P2 < I , in a situation characteristic of the non-adiabatic domain .

b Activation energy of small polaron at high temperatures Here we will establish the expression for the activation energy characteristic of a high temperature regime, before exploiting these results in Section 4 in conjunction

V Conduction in delocalised, localised and polaronicstates

121

I-I I I I

(b)

I I I I

(c)

I I I I I

1

I I I

Figure V-IO. Hoppingprocess for small polaron: (a) the dot represents the positionof a carrier on an atomic site; (b) coincidenceevent is formed; (c) carrier's situation followingits transfer and relaxationof surrounding lattice.

with the knowledge that the minimum energy required for a 'coincidence event' is WH ~ -Ep / 2 to derive calculations for conductivity. Only with the energies of two wells (well for departure and for arrival) being equal can we enter the configuration shown in Figure V-9 C2 (coincidence event). Using the notations given in Chapter IV, Section II-3, we can write this as -Axi = -AX2, or xi =X2. In going from Figure V-9-a to Figure V-9-Cl the well s from which the charge originates must be deformed with an energy B(x] - xO)2 in which xo and x] are, respectively, the deformation of the wells in their initial state (polaronic state associated with a deformation Xo calculated in Chapter IV, Section II-3-b) and in the state cj , The well which ' receives' the charge undergoes a change from nan-deformed configuration (non-polaronic wells as in the initial stage shown in Figure V-9-a) ta a deformed configuration X2 (Figure V-9-C2), which necessitates an energy BX~. The nece ssary energy overall is therefore W = B(x) - xO)2 + BX~, and with Xl = W = B(X2 - xO)2 + BX ~. W , the minimum energy and denoted WH required for a coincidence event can be obtained by minimising W with respect to X2: aw . Bx 2 as aX2 = 4Bx2 - 2Bxo = 0 and thu s X2 = xo/2 = x i , we obtain WH = and then

T'

with Ep = -Bx5 (see Chapter IV, Section II-3 or II-4), we finally have WH = -

E{.

c Comments a Comment J When the two wells of departure and arrival of the charge are energetically separated by t!.. at the coincidence event due to disorder (Figure V-9-c), the original equation (- Ax ] = -AX2) should now be replaced by A(xi - X2) = t!.. . This result s in [Mot 79] : Ep t!.. WH= - - ± - .

2

2

122 Optoelectronics of molecules and polymers ~

Comment 2 When the material under study is polar, we are dealing with large

polarons and the expression that we have just established, WH = - ~ , is acceptable only when the localised charge carrier on one site does not modify the distortion of another molecular site, i.e. as is found for small polarons. For a polar material, therefore , this expression in no longer applicable, as Coulombic potential wells overlap between the two sites and mutually perturbed. y Comment 3

The transfer probability for a small polaron at low temperatures is

given by C exp ( -

~: )

exp (-

~ ) (page 81 of [Mot 79]), is valid for a temperatures

such that kT < [(l /4)hw] and gives rise to non-thermally activated processes when /:!,. ~ O. For intermediate temperatures, the relationships obtained for high temperatures can be reasonably used, with the condition imposed that we replace WH by

r, WH ( 1-"31 liw)2 kT .WhenT = Te 2 ,energydecreasesby8 %andwhenT = 4,by30 %. 4 Relationships for continuous conductivity through polaron transport We have seen, starting with our study of small polarons in a crystalline system, that the presence of disorder in a non-crystalline system localises charge and slows transfer, thus aiding the formation of small polarons. In order to follow transport phenomena associated with small polarons, we are compelled to consider relation ships for continuous conductivity (and if possible thermoelectric power, as detailed in Appendix A-6) with respect to temperature. Depending on the temperature range envisaged, in Section 2 we saw in Figure V-9 that three transport mechanisms are possible:

a Very low temperatures (kT < 10-4 eV) At very low temperatures, the only process likely to exist is that shown in Figure V-9-a and reproduced in Figure V-ll -a below. This process occurs without changing the overall population of phonons and for a charge subject to a double tunnelling effect with a deformation associated with the movement of a polaron within its own, extremely narrow band. For a cubic crystal, the band is of size B = 12 J exp( -S) and the effective mass can be written using eqn (22) from Chapter II, i.e. m* =

~ a

=

J liz

2aZ exp(-S)

. The corresponding mobility is in the form It = ~~ 2a 2 J exp( -S),

is proportional to exp( -S) and is extremely low, much as the corresponding conductivity. Aside, it is probably worth considering whether or not this process occurs in amorphous semi-conductors [Ell 90], given that fluctuations in tail energy band sites are perhaps higher at 10- 4 eV.

b At low temperatures At low temperatures, although not as low as those in (a), we have the process detailed in Figure V-9-b which consists of a charge subject to a double tunnelling effect and a

V Conduction in delocalised, localised and polaronic states

(a)

(b)

123

(c)

Figure V-U. Transport mechanisms at different temperatures. (a) Low T; (b) Intermediate T; (c) 'Elevated' T.

vibration assisted by phonon overlapping. The activation energy of this proce ss decrea ses with temperature (and tend s towards zero for condu ction in the polaronic band ). A variable distance hopp ing mech anism with these characteristics is probable [Gre 73] followin g a law of the form Lno ex: - (T a/T ) 1/4.

c 'Elevated'temperatures At slightly elevated temperatures, around or ju st above ambient temperature, the process detailed in Figure V-ll-c may occur, requiring a single tunnelling effect coupled with the 'event coincidence' due to the same vibrational configuration of two adjacent sites resultin g from a therm al energy WHo Here we can write that a = qN IIp, in which N is the charge carrier density and IIp is the mobil ity associated with this type of polaron. On using Einstein' s relation ship , IIp = qD /kT (with D in the form D = Pa 2, in which P is the probability that a carrier will hop between neighb ouring sites and a is the inter-atomic distance) we end up with:

Nq 2a2 a = - - P. kT P depend s on two probabilit ies which were defined in Section 3 of this Chapter:

•

•

PI is the probabilit y for the event coincidence in the form PI = va exp(- WH/kT) in which va is the mean phonon frequency and WHis the activation energy required to produce the equivalent of the two sites. As we have seen in Section 3-b, WH = - Ep / 2, and from Section 3-c, disorder can be represented using the param eter ~ for the amplitude of differences between initially occ upied and unoccupied sites. We can therefore now say that WH = -Ep / 2 ± ~ / 2 with Ep repre senting, as previously denoted, the polaron bonding energy - A2/2Mw6 for an isolated diatomic mole cule . Thi s activation energ y can , however, be reduced as the lattice slowly relaxe s, leading to a correlation between successive hops. P2 is the probability of charge transfer during an event coincide nce. For the nonadiabatic regime , in which P2 < I as carriers are slower than lattice movement s, the relati onship 1/ 2 I 11 2 J nva ( WHkT ) was obtained by Holstein [Zup 9 1] in which J is, as before, the electro nic transfer integral such that J2 = exp(- 2aR). P ~= -

124 Optoelectronics of molecules and polymers Under non-adiabatic conditions, the mobility can be written as: 2

[L

2

qa ( -II- ) qa = -P\P2 = -hkT kT WHkT

J2 exp (WH) - - -----+ I-l ex T- 3/2 exp (WH) - . kT

kT

This mobility appears thermally activated over a wide range of temperatures and characterises polaron based conduction. When the temperature is such that WH is of the order of kT, the preexponential term predominates and gives rise to a variation proportional to T- 3/2 . We also note that N = N, exp( -Eo /k'I'), in which Eo represents the activation energy necessary for a constant number of carriers, corresponding to an equilibrium value, and N, (at approximately 1022 cm- 3 is a value higher than that generally seen in non-polaronic theories [Ell 90]) represents the density of equivalent sites initially susceptible to thermally generate carriers on polaronic levels. We thus arrive finally at an expression for conductivity at 'elevated' temperatures: 2

° = qakT N, VOP2 exp (

Eo+WH) kT

= 00 exp ( -

Eo+WH) kT

In general, 00 ~ 10 - 103 [2 - 1 em:" .

Comment 1 For a one dimensional model, in which the transfer of vibrational energy giving rise to equivalent sites depends only on one direction we have WH ~ -Ep /2. However, with a 3 dimensional model , supposing that energies are propagated equally in all 3 directions, for each direction we have WH = -Ep/6. Comment 2 It should be noted that for intrinsic carriers, Eo is less than half of the optical gap. (The optical gap corresponds to an excitation of an electron from the valence band to the conduction band, during which atoms remain unmoved in a rigid lattice) . However, when atoms can adjust their position in response to the presence of a carrier, there is a coupling between charge and lattice consisting of small polarons and the energy scheme for this is illustrated in the right hand side of Figure V-12-an extension of Figure V-8. The size of the gap can be deduced from the probability of a level being occupied by either a small negative (localised electron) or positive (localised hole) polaron with the energy difference between them being 2 Eo. With the bands of small polarons being narrow, (kT » polaron band size) the Fermi level can be found midway between the bands of electrons and small polaron holes in an intrin sic semiconductor.

5 Conduction in 3-D in n-conjugated polymers It should be said that most calculations concerning n-conjugated polymers have been performed on single chains, without calling into account interchain interactions. And yet, isotropic conductivity is often observed for doped n-conjugated polymers, tending to indicate that interchain charge transport is relatively facile due to the presence of a non-inconsequential overlapping between n-orbitals of adjacent chains . Models of conduction have been elaborated for this scenario, given that electronic transfer between chains is effected either by direct three dimensional coupling or via intermediate doping ions.

V Conduction in delocalised, localised and polaronic states

125

/ / / / // / // / / / / / / / ///,

I : Thermalenergy enerating carrierson laron levels

Activation ene y necessary for a one-directional hop, corresponding to conduction by hopping potarons with predominant'non diagonal' process

Smallpolaron narrow band associated with generally egligible diagona process

------j

I I I I

Figure V-12. Energy scheme for charge-lattice coupled system (intrinsic charge generation).

a Direct 3-D coupling between chain s Direct, 3-D, interchain coupling is charac terise d by the transfer integral t.L (Fig ure V-13). In order to perform direct calculations of the band struct ure in 3-D, we need to know the actual 3-D structure under co nsideration. While, in general, only rela tively inco mplete data are available, for polyacetylene (PA) and poly(pa ra -phenylene vinylene) (PPV) local determination s of t.L integrals have been made yielding values of the order of 0.4 e V [Gom 93]. Such a value ass ures for polyme rs like PPY the condition (t.L/t/ / ) ::: 10- 2 , a value generally considered necessary for 3-D transport [Sch 94 ]. Indeed , it is worth noting here that such transport ca n be established without resort ing to the intervention of polaro ns. For PA and PPV apparently reasonable method s have been used to eval uate interchain interac tion effects on polaron stabilisatio n. Using a model identica l to that derived for isolated chains, it has been show n that polaron effec ts are considera bly weakened once interchain interactio ns are taken into acco unt: reasonable value s for 3-D coupling were sufficient to destabili se polarons so that they could no longer practically form .

1

t.L

(

Polymer chain 1

1

- _

Polymer chain 2 Figure V-13. lntrachain (t / /) or interchain (t.L) transfer integrals.

126 Optoelectronics of molecules and polymers

OuPuO-o. t 1' . dopant d~

OvOoVo til

I

h

"

Figure V-14. Schematisation of dopant bridging showing various transfer integrals.

All the previously mentioned calculations were actually performed using an ideal structure, without geometrical or chemical defaults, and as we have already seen in Chapter III, Section IV, it is disorder, for example due to certain short conjugated sequences, or chemical defaults such as polymer precursors remaining in the material and dopants, which tend to stabilise polarons [Con 97] .

b Chain coupling through intermediate dopant ions [Bus 94] Interchain coupling has been , in particular, studied for (t.l/t/ / ) 2: 10- 2 not being true , using , for example pol ypyrrole in which t/ / ~ 2.5 eV and t.l ~ 0.03 e V. The presence of a dopant ion inbetween chain s which can transfer carriers with equal ease to either one chain or the other-as shown in Figure V-I4-can assure coupling; the resulting transfer integral td ~ 0.5 to I eV is significantly larger than t.l . The electronic charge transferred by dopant or 'counter-ion ' is distributed on the chains adjacent to the ion. As a result, polaronic species called tran sverse polarons and bipolarons form acro ss several chains and are centred about the counter-ions (Figure V-IS ). Calculations have shown that transverse polarons can only exist when the dopants are well disper sed from each other and that at higher doping levels, tran sverse bipolarons form . Both the attractive potential induced by the dopant and its tenden cy to rein force tunnelling effects stabilise transver sal polarons and bipolarons, as shown in Figure V-16.

Polymer chain

• Polymer chain (a)

• v'" 4·

·u·

-J===========:::t(b)

Figure V-IS. (a) Transverse polaron; and (b) transverse bipolaron. v is the polaron half size expressed in monomer units.

V Conduction in delocalised, localisedand polaronic states

127

~ Dopant centers reinforce transitions between two chains due to tunnelling effects

Figure V-16. Reinforcing interchain transitionswith doping centres.

c Hopping conduction in disordered polymers On a small scale, when there is a high enough concentration of dopants to connect adjacent chains, then clusters of polarons resembling networks form . We can see though that on a larger scale, there is an inhomogeneous spread of polarons in a disordered polymer network, and the polaronic clusters are separated. A calculation for conductivity by hopping for this configuration has been made using the following, three hypotheses: • • •

adiabatic hops occur within polaronic clusters; non-adiabatic hops occur between polaronic clusters, with the intervention of a phonon; electrostatic energy is the principal barrier in the hopping mechanism. The relationship found for conductivity is [Zup 93]:

Here To depends on both the charge energy of clusters and system granularity such that

where

I

e2

4TI£O £r

a

u=--and S is and and

represents electrostatic repulsion between two electrons separated by distance a, the lean distance between dopants in a cluster and is independent of cluster size & is the mean distance between dopants assuming a homogeneous distribution ignoring clusters. On finally using the representation of a simple inhomogeneous distribution of dopants, supposing that there are no particulates with metallic properties, the obtained law resembles closely that of Sheng (see below), which was established for a granular, metallic system.

128

Optoelectronics of molecules and polymers

VI Other envisaged transport mechanisms This Section contains a relatively short summary of the different models, and their laws, used to interpret the behaviour of conductivity in organic solids.

1 Sheng's granular metal model Sheng's model was formed in order to comprehend the observed conductivity of hetero geneous systems composed of nano-particles within an insulating ceramic matrix and supposes that the metall ic clusters have free charges which can pass throu gh the matrix followin g a tunnell ing effect. The charge transfer is limited by the energ y E Ch necessary for an electron to hop from one particle to the next, which will form a positive charge on the particle it has left and a negative charge on the particle to which it has arrived. Using the idea of a charged cap acitor as a reference, we can suppose that the system will have a charging energy. The law obtained is in the form

with To dependent on Ech, the distance between grain s and the transparency of the barrier between grain s.

2 Efros-Shklovskii's model from Coulombic effects In Mott' s model of variable distance hoppi ng, it is supposed that charges have negligible interactions between each other. When Coulombic interactions between a charge and a remnant positive hole becom e dominant (with openin g of a Coulombic band gap as deta iled in Chapter II, Section IV-2), the mean hoppin g distance can be evaluated to give, in all dimen sions, a law of the type: In p ex (TofT) 1/ 2,

in which a = ao exp(- T ofT ) 1/ 2.

Sheng also showed that this law resembled his own, with the Coulombi c repulsions simply playing the role of charge energy.

3 Conduction by hopping from site to site in a percolation pathway Conduction along a percolation pathwa y within a lattice is modelled. It is supposed that the conductivity a = GmfR m, where G m is the characteristic conductivity and R m is the characteristic dimension of the lattice. The percolation limit is assumed to be reached once the limitin g value of G m is equal to Gc which is in tum defined as the greatest co nductance permitting a continuous passage throughout the lattice for all charge movement s (Gij) between sites i and j which are such that Gij > Gc. The obtained law, which resembles that of VRH, is a = ao exp( _ AT - 1/ 4 ) .

V Conduction in delocalised, localised and polaronic states

(a)

129

(b)

Figure V-17. Heterogeneous systems with: (a) moderate dopant concentration; and (b) high concentration.

4 Kaiser's model for conduction in a heterogeneous structure Conductivity in a heterogeneous structure was modelled by Kaiser [Kai 89] in order to account for the behaviour of n-conjugated polymers doped to varying degrees. Here we shall detail only expressions for continuous conductivity, although those for thermoelectric power can be followed in Appendix A-6. In order to understand a variety of behaviours, models of heterogeneous polymers with 2 or 3 domains were proposed [Mol 98]. As shown in Figure V-17-a, the former system was considered to consist of fibrils (Region I) within a matrix (Region 2) with the conductivity such that 0'-1 = f l0'1 1 + f20'2 1 where fj being a factor in the form fj = LjA/pLAi with p representing the number of conducting pathways given here by the same ratio ~~ of length over sectional area (L and A are the sample length and cross sectional area respectively) . 0'2 due to the presence of sufficient dopants, we have 0' ~ f210'2 and When 0'1 the evolution of 0' is pretty much guided by the term 0'2 . This means that transport mechanisms assisted by tunnelling effects are dominant, while at low temperatures the hopping mechanism (VRH) proposed by Mott are of relevancy. It should be noted that 0' can be high due to the term f21 which is proportional to L/L2 with L being high relative to L2, the thin inter-fibril barrier). At high levels of dopant, in order to account for a finite value for 0' at temperatures tending to 0 K and the increase in thermoelectric power with T at low temperatures, the preceding representation using fibrils needs to be modified as shown in Figure V-I?-b. Two parallel domains are substituted, the first of which (conductivity 0'3) continues to represent inter-fibril hopping, while the second introduces, like an amorphous metal, a supplementary component (0'4) in the form 0'4(T) = 0'40 + aT I/ 2, in which 0'40 and a are constants. Overall , this gives 0'-1 = (fIO'd - 1 + (g30'3 + g40'4)-I .

»

VII Conclusion: real behaviour of conducting polymers and the parameter w = -d(log p)jd(logT) 1 A practical guide to conducting polymers At the level of the actual, practical behaviour of polymers, an article by J P Travers [Tra 00] is of considerable use as it details magnetic characterisations of different molecular structures which may be used to determine charge origins. Here though we cannot focus only on electrical transport, and can at best give a summary of the

130 Optoelectronics of molecules and polymers

a /a(300)

Low doping level

( t1E+WH ) o= croexpC kT orVRH

o

300 T(K) Figure V-18. Conductivity of doped CPs at different temperatures.

three principal classes of conducting polymer s (CP) with respect to the conductivities shown in Figure V-18:

a Highly doped CPs The se polymer s exhibit a 'm etallic' character (degenerate medium with EF in delocalised states due to a disordered system). Conductivity parallel to the chain axis can be expressed using Drud e-Boltzmann 's relation ship (Sec tion III-I-a):

Given that kF = ~ , we obtain q 2a 2

_(n rrn

a/ /- - - )

(£) a

(equation retained by Kivelson and Heeger [Kiv 88]. With respect to temperature: •

at low temperatures with poor locali sations, we have seen from Altshuler and Aronov's law that

•

at slightly higher temp eratures, inela stic colli sion s are controlled by electronelect ron interaction s, with I <X T - 1, resulting in a <X T (Mott's linear law); and at higher temperatures, collisions with phonons dominate with I <X T - 1/ 2 , again giving (J <X T 1/ 2 in a law identi cal to that propo sed by Kaiser for highl y doped hetero geneous systems.

•

V Conduction in delocalised, localised and polaronic states

131

b Moderately doped CPs For polymers with a moderate level of doping, we can suppose that:

•

•

EF tends to be outside extended states, with states more or less localised within the gap. There are different theories, however, they result in laws essentially following o ex: exp(-To /T)x. there is a heterogeneous structure [Kai 89J, with conducting fibrils separated by an electric barrier: cr- I = flcrl l + f2cr21 ::::} (and with o l »cr2, cr ~ f2"lcr2 --+ VRH (tunnelling effect) again in the form o ex: exp( - To/T)x.

c CPs with low levels of doping Here, EF is within a quite large gap of few states . Whatever our point of view concerning polaronic states or states tied to localised defaults, we keep coming back to a VRH based law at low temperatures, while at higher temperatures a thermally activated law is more appropriate.

2 Temperature dependence analysed using the parameter w = -[(a In p)/a In T] We can suppose that in the majority of cases, resistivity can be expressed using the general formula [Zab 84] p = BT-mexp(To /TY . We can go on to derive In p = In B - mIn T dlnp _

- T dT - m

+ (~o )x, giving

_ + x (To)X T ' and w - -

also d~~p =

dlnp _

d In T -

m

-!f - X T~il . Finally this yields

+ x (To)X T .

a In the VRH regime In this domain, we have m

= 0 and x = 1/4 in 3-D, with w = ~:t ( )

1/ 4

.Following

the preceding law, we have In w = In(xT6) - x In T = A - x In T. In Figure V-19, the curve representing In w = f(ln T) in an insulator-semiconducting domain shows it to be a straight line with a negative slope -x = -0.2. We can also note that (To/T)X = (ToTI -x) /T = E(T)/T so that: • •

when x = 1, E(T) = To = constant, and we are in a thermally activated regime with p = po exp(~E/kT) in which ~E = kTo; and when x = 1/4, E(T) = (ToT 3 / 4 ) which is such that E(T) decreases with T in a behaviour characteristic of VRH i.e. activation energy kE(T) decreasing with T.

b Close to metal-insulator transition Resistivity is not thermally activated and p(T) ~ T"?". Thus we have To = 0, with In w = In m = constant. The representation In w = f(ln T) is a straight horizontal

132

Optoelectronics of molecules and polymers W

=-d(ln p)/d (ln n

T(K)

Figure V·19. Ln curves for wfI') as a function of temperature for PANI doped with camphorsulfonic acid under three regimes: metallic, critical (M-I) and insulator.

line , as shown in Figure V-19. This law corresponds to the TCR (temperature coefficient ratio) up = (l/p)(8p/8T) < 0, while a real metallic character follows up> O.

c True metallic behaviour True metallic behaviour could be observed with polyaniline (Pani) [Men 93] with peT) increasing very slightly with T at low temperatures with a system changing from a characteristic power law under critical regime (close to the M-I transition) to purely metallic behaviour.

VI Electron transport properties: II. Transport and injection mechanisms in resistive media

I Introduction In the preceding Chapters , we have attempted to define the structure and nature of electronic levels following charge insertion into organic materials, in no particular order, by doping, electron injection or photoexcitation . In organic electroluminescent diodes , doping agents need to be removed as they can quench luminescence. Given that the materials in these types of components are more like insulator s than conductors , it is useful to understand: • •

electron and hole injection at the cathode and the anode, respectively; and transport of electrons and holes at the electrode interface or within the organic layer.

As we shall see, different mechanisms can be envisaged. The I(V) characteristics (current - voltage characteri stics) of an electronic structure are often studied and explained in terms of charge injection either under field (Schottky) or tunnelling effects. Alternatively, results can be interpreted using models based on current flow limited by space charges (SCL). The aim is to reduce the threshold potential of a diode, and depending on the dominant process, it should be possible to improve the electronic properties of the metal used for the electrode or the mobility of carriers within the organic layer. In this Chapter, we shall present the various possible charge mechanisms for charge injection at interfaces and study how, classically, electron transport is envisaged in near-insulating materials such as undoped organic solids, which display conductivities of the order of IO-IOQ-1cm - 1. An example of which is well purified poly(para-phenylene) (PPP), which exhibits conductivit y around 1O-1 5Q- 1cm-1.

II Basic mechanisms 1 Injection levels In order to attain significant electric currents within components, such as electroluminescent diodes, charge carriers are required to be injected in a high enough

134

Optoelectronics of molecules and polymers

co ncentra tio n to en ergeti call y high level s using a sufficie ntly inte nse elec tric field (E a) . Th e initial thermal eq ui librium shown in Figure VI-l-a is brok en , givi ng rise to thermall y excited carriers at the outer edges of co nduc tion bands (C B) (in the case of e lectrons) and a pseudo-equilibrated regim e (F ig ure VI-I -b ). Th e Fermi leve l (EF) yie lds Fermi pseudo-level s fo r electrons (EFn) and holes (EFp) wi th a co nsiderab le increase in charge concentra tio n [Mat 96].

2 Three basic mechanisms Cl assicall y, outside of eq uilibri um, three typ es of current ca n be co nside red : (A) At -

(a)

e lec tro des, the curre nt of ca rriers is produced by: thermoelectronic e mission; emiss io n due to field effec ts (Sc ho ttky); and emission due to tunnelling effects;

(h) --.~n 'ection level (» lower than e B) C t R Thermal rclaxationrrs « t v)

CB

0

= n exp ( E C -E I. ) ' kT Carriers genera ted in the band bottom permitted by thermod ynamic equilibrium

v voo

Fn

E,

v

= n , exp

(E-E) Co

kT

I

Thermalised carriers in the bottom of the permit ted band

V.

Cathode

o

E

Anode

d

Figure VI-I. (a) Thermodynamic equilibrium (EF, no) If the applied field (Ea) is weak, carrier concentration in permitted bands is unmodified and there is no injected space charge (p = 0). Conduction is assured only by internal charges, of concentrations no and PO, at thermodynamic equilibrium. For electrons, In = q nO Iln Ea. Integration of Poisson's equation, b.V = 0, with limiting conditions, YeO) = 0, V(d) = Va, gives - ~ = E(x) = Constant = Ea = - Va/ d, that is: Vex) = (x/ d)V a (ohmic regime). (b) Outside thermodynamic equilibrium (EFn, n) Charge carrier concentration in bands is modified by Ea (the mean applied electric field, as defined by Ea = - Va/d, in which Va is the applied field at the insulator/semiconductor). Tied to carrier injection and the high resistivity of the material is space charge with p i- O. Ea is sufficiently high to introduce by emission or injection from electrodes concentration of carriers breaking down the initial thermodynamic equilibrium. The current, J, being governed by intrinsic properties of material, is no longer ohmic and is either: - limited by trickling properties of charges introduced in the material volume (l Ev); or - limited by current produced at electrodes (le) . 1 = lEv if l Ev < l C; 1 = l C if l C < l Ev.

VI Electron transport properties

135

Figure VI-2. Three basic mechanisms A, B and C. (B) Current of a single carrier type at one electrode limited by the space charge in the electrode locality CJsed . In the presence of trappin g levels in the insulating volume , the space charge limiting law must be modified to account for reduced mobility of carriers; and (C) Current due to doub le charge injection at each electrode controlled by the insulating volume . Current is limited by space charges (or electrode currents as they limit injection) and a concentration of recombination phenomena. In the following Section we shall look at each of these currents in more detail.

III Process A: various (emission) currents produced by electrodes Typically these currents occur when the metallic electrode and insulator contact is rectifying (if W M and W SI are, respectively, work functions for the electrodes and the insulator or semiconductor, when W M > W SI) . Depend ing on the strength of the applied electric field (Ea), three types of electric current can be identified and are detailed below.

1 Rectifying contact (blocking meta l - insulator) Figure s VI-3-a and b show the rectifying contact with W M > W SI. Under these condition s, on contact electrons are emptied into the metal (M) from the insulator (I). In I there appears a positive space charge over a relative large distance (L) due to the low electron density in I. This is called the depletion zone.

Va 0 < 0 SC side (direct polarity) > 0 SC side (inverse polarit y)

M side barrier

lor SC side barrier

WB WB

qVd qVd - qV a

WB

qVct

+ qV a

Result vis a vis electron passage

I or SC - M eased M """""* I or SC remain s difficult I or SC """""* M difficult M """""* I or SC always difficult

136

Optoelectronics of molecules and polymers

Vacuum level

E I-_ ---K._ ..:.:•·E,

'Cur

Metal

in. ulator

letal

(a) before contact

(b) after contact, with effect of

V appll
<0

Figure VI-3. Metal-insulator contact with WM > WSI. The Table above summaris es characteristics for a contact with WM > WSI which is effectively blocking with respect to electron injection (M ---+ I). Only a thermoelectronic emission can occur from M (which tend s to saturate in the form JOSCH A*T 2 exp (- [WB - t>. W] /kT due to emissi on by field effects). With a high value of E, a tunnelling effect (generated with a narrow barrier) can allow thermoelectric emi ssion through this blocking contact (Fowler- Nordheim equation ).

=

2 Thermoelectronic emission (T ::j:. 0; E a = 0) The Dushman-Richardson law (deduced from distribution of electron velocities in metals given by Maxwell-Boltzmann's equation in Appendix A-7 ) covers the emi ssion process detai led in Figure VI-4 . The emitting current density at saturation is given by Josr = A*T 2e xp (-WB / kT), with WB = WM - X. in which WM is the metal work function , X. is electron affinity of insulator or semiconducto r and A* is the modified Richardson constant from A* = (4Jt qm*k 2)/ h 3 .

3 Field effect emission (Shottky): Ea is 'medium intense ' In this process, the electri c field (E,) reduces the potential barrier by t>. W = q(qEa /4m;) 1/2 . The law used in Process I to give the saturation current here becomes Potenti al energy x

en Ictal W, E:L-

W" f X -l

- - - EC- --- -- ·

Insulat or or

S

Figure VI-4. Emission process by thermoelectric effect.

VI Electron transport properties

137

Potential energy

x hLmlator or S Metlll

---lfc- - - - - - -

E. Figure VI-5. Emission process due to Schottky field effect.

= A*T2 exp( -[WB -

!':1 WJ/kT) (as in Figure VI-5 and Appendix A-7). With T constant, theoretically log JOSCH = f(JEa ) remains linear, however, at low E a , J is often observed to depart from a straight Schottky line due to the appearance of space charge near the emitting surface (see Figure VI-21) . JOSCH

4 Thnnelling effect emissions and Fowler-Nordheim's equation At high values of Ea , the potential energy curves steepens and the potential barrier observed by electrons narrows considerably, only to be crossed through Process 3, which corresponds to a tunnelling effect (Figure VI-6 and Appendix A-7) . The current density is given by Fowler-Nordheim's equation:

E; [4(2m)

3

JOFN

q = 8JthWB exp

- 3

-

1/2

fl,2

W~2]

-- . qE a

Log(JoFN /E;) = f(I /Ea ) is a straight line with slope WB. Potential ene r gy

- qE. x

Metal

Insulator or SC ncuum level

lnjection le vel

»

ground of BC

~

Figure VI·6. Tunnelling effect emission process.

138 Optoelectronics of molecules and polymers

IV Process B (simple injection): ohmic contact and current limited by space charge Current limited by space charge occurs when the contact is ohmic, that is to say that the electrode behaves as though it is an infinite reserve of charge and the current flow is not determined by the volume of the emitting material. Given that the latter is close to being an insulator, the formed space charge opposes the trickle of current through the material within the neighbourhood of the electrode-material interface. At a sufficiently high applied field (Ea), space charges are pressed right back to the interface and can effectively enforce a saturation current. Here, we shall look at the different stages involved.

1 Ohmic contact (electron injection) a Definition A metal-insulator (M - I) contact is ohmic when its resistance (impedance) is negligible compared to that of the insulator or semiconductor volume. As a consequence, free carrier density near the contact is considerably greater than that in the material volume, which is generated thermally, and ohmic contact can act as a charge reservoir. The conduction is controlled and limited by the insulator volume impedance and any possible recombination phenomena.

b Comment The use of the term 'ohmic' is not particularly exact. Under high fields the concentration of charge carriers rapidly overshoots the concentration no, which gives rise to the ohmic law generated intrinsically in the material volume . However, with increasing field strength, we can find that n rapidly exceeds no due to inequilibrate charge injection, resulting in the I(V) law becoming non-linear and thus non-ohmic .

c Realisation and schematisation Ohmic contact is obtained when WM < WSI. As shown in Figure VI-7, once contact is attained, equilibration of Fermi levels (EF) generates a negative space charge (p) on the side of the insulator which extends over a narrow accumulation zone-due to carrier accepting state densities being large, N, ~ 10 19 cm- 3-and thus of low resistance. The result is a curve in bands with a reduction in the M -+ I barrier, such that it equals WM - x.. If (WM - x.) ~ 0 or is negative, then the contact is ohmic . With the application of E, in the sense shown in Figure VI-8, i.e. directed from the I to M and thus with Va > 0 at I, it is the whole volume of the very resistive I which suffers the drop in potential, in a drop which decreases with Va. On taking into account the antagonistic effect of Einternal (due to p and directed M -+ I) and E a, the conduction

VI Electron transport properties

139

I.

E

- I---EE·· . ·_. ·-~~~~ .J i

Metlll E..

r: . \4' -, ....~.~~~:~J{

\ \j; L_ SC .~ .

1- __

E..

J

i i.

~ !...........~~

E vacuum

;.:

(a) Befor e contact (WS I > W .\I )

-Mo-=-;(- - - X :\lct:t1 C\l1

1(lr C

~~--Y E, Arter contac t (curve a mplitude is

E\' (h)

r = W SI _

W'I )

Figure VI-7. Metal-insulator contact with WM < WSI .

band in I (denoted by Ec in Figure VI-8) goes through a maximum at x = Le , at which point dV Idx is zero, as is Eresulting (V being the resulting potential). Therefore, the two effects of internal and external fields compensate one another exactly at x = Le , a point which is called the 'virtual cathode' . As Ea reaches (V I -+ V2 with V2 » V I), L, -+ L~ with L~ < Le ; the virtual cathode tends towards zero and space charge at this point is repressed. The result is that current can more easily pass through I to a value which is the saturation current limited by space charge JSCL .

2 The space charge limited current law and saturation current (Je) for simple injection in insulator without traps a Hypotheses for simple cathodic injection of electrons I) A band model is applicable to the treatment of the injection of carriers with a current non-limited by an electrode (for example with perfect ohmic contact).

..

E lnlrm"" .I _ _•

I

•

Vir tual cathode

E vacuum

x

E.. '

i\letal ( i\I ) <,

I I: Insulator

,,

I

E

,

.C1

........ ....

<,

",En ' .

Figure VI-S. Effect of applied field (E a) in which polarisation Va > 0 on insulator, with Va = V 1 or V2 such that V2 > V I .

140

Optoelectronics of molecules and polymers

Considerable controversy surroundedwhether or not band models, necessaryfor the description of an ohmic contact, could be applied to organic solids.A model does indeed exist for organic solids [Pfu 86], [Mol 96 and 98] (for polymers, see Figure VI-8) and is close to that of amorphoussemiconductors. It should be said that even in inorganicsemiconductors, a representation of continuousbands is an approximation. The most important point remains the existence of energy levels within an insulator that can accept, at the interface, injected charges, whether those levelsare localisedor not. Either way, effects on the resultingmobilities are taken into account because this parameter is included in the final expression for current density. 2) Carrier mobilities are independent of Ea and dielectric permittivity (E) is NOT modified by charge injection. Poole-Frenkel or impact ionisation effects are not considered. 3) Electric field is assumed high enough to consider the following current components negligible: • current due to thermallygeneratedcarriers with density no: ohmiccurrentdue to these carriers is negligibleunder a high injectionregime as n » no; • current due to diffusion. Here we can note that the appliedpotentialis considerably higher than the thermal potential kT/ q. The derived term for current (conduction) is dominant with respect to that of diffusion: 1 = crE+qD n grad n

= qn(x)I-LnE(x) + q(kT/ q)I-Ln(dn/ dx) = I-Lnp\(x)E(x) + I-Ln(kT/q)(dp\(x)/dx)

~

I-Lnp\(x)E(x) .

with the form found a posteriori for p,(x) ~ x- m (and m ~ I : see following Section c) we have: Comment:

Idiff Iderived

kT exE(x)

kT eV(x)

--~--=--

at Tambient we have kT = 0.025eV; with Vex) » 0.025V the approximation is justified. 4) Weconsider that the system is one dimensionalwith two planes, the first at x = 0 for the cathode, injecting electrons, and the second at x = d for the collecting anode. Ea is sufficiently intensefor the resultingfield at the cathodeto be zero (the sameforbothreal and virtualcathodes); E(x = 0) = -(dV/dx)x=o = O. The current in this situation is the saturationcurrent (Js) following Mott's approximation and is the maximumcurrent which can transversethe insulator.

b Mott-Gurney's expression (1940)

Preamble: in a stationary regime, derivative of j = 0, or rather 1 = constant with respect to x, such that 1 = qn(x)I-LE(x) . Thus, to realise Is, we just need to calculate it with respect to x = d.

VI Electron transport properties

141

Form of E(x) : Poisson's equation gives dE (x)

p(x)

dx

E

1

qn (x) E.

Ej.1E(x)

By integrating 2E( x) dE(x) = d[E (x)]2 = 2J and using limiting co nditions (E( O) = dx dx q.l ' for 1 = 1s ) we have:

o

2 ] 1/2 E(x) = - [ Ej.1 1s x

(1)

(Notice the negative sign which has been used as Va > 0 and the mean field (E a) is r d -dV = -Va = JrdO E (x)d x = Ead, or E, = - -t). v defined by: Jo •

Expre ssion for 1s : here d lo

- dV = -Va = l d E(x)dx = -

(21 )1/2 Ej.1

0

d

1

_ s

0

(21 ) 2

x l / 2d x = _ _s 1/2 _d 3 / 2 , £j.1 3

and leads to 9 V2 9 E2 1 = - Ej.12 = - Ej.12. s 8 d3 8 d

(2)

c Graphical representation as in Figure VI-9 Placing eqn (2) into eqn ( I) gives

E(x) =

3 v, (X) - 2d d

=

3

(X) I/2'

2Ea d

and E(d) = ~ Ea . Additionally, Vex) = -

1r

E(x )dx = -Ead

(X)3 d /2'

0

and thus Vex) = V a ( j )3/2. The potential energy W(x ) of an electron can be directly dedu ced from Vex) as W (x) = -qV (x) (Figure VI-9-b insert). The charge density (p) has the form p(x) =

dE 3EEa (X )- 1/2 = . dx 4d d

£-

Comm ent J We can see that the empirically derived curve of the law for yield determination of electron mobility u .

1SCL

can

142

Optoelectronics of molecules and pol ymers

Comment 2 Here we consider up to what frequency the static characteristic Js = f(V a ) can be applied . So that eqn (2), for a stationary regime (derivative j = 0), can be applied to a dynamic regime, the transit time of an electron (ts ) through thickness d must be less than the period (T) of the applied electric signal i.e. t, < T = 1[». In other terms, the critical frequency (v e ) of the applied field cannot go higher than the point at which t s = l ive · With dt = Q!'. in which Yn = -fLnE, we have V o

ts =

f

dt =

f

d

ioo

_ _d_x_

fLnE(X)'

F d Position of virtual cathode when E(O) '# O. J '# Js,

I

E(x)

----..;.:~==t

CLE" (x/d)ln I =(3/21

IE(d) =3/2 E I ~

W (x )

o

v

p = 0; J = 0; V(x ) = (x/d) Va

v L Va (x) = (x/dr Vl

I

d

p

o

d

I-J=- 0-:p-=-OI

Figure VI-9. Representations (a) E(x) ; (b) V(x) , W(x) and (c) of p(x) ; J constant is observed as if p(x) is small, E(x) is large and vice versa.

= aE = fLP(x)E(x) =

VI Electron transport properties

143

following eqn (I)

E(x)

=-

2 ] 1/2 , [ €IlJsx

and therefore, t = s

[_€_]1 /2 21lJs

[d x- 1/2dx =

10

Taking into account eqn (2), we obtain t, =

*~

11 a

[~]1 /2 d 1/2. ur,

and the relationship ts < I [» gives

I 31lVa v < v = - = --. c ts 4 d

(3)

In terms of actual numbers, when Va = 5 V and d = 111m, we obtain

Vc ~

60GHz . 3 Transitions between regimes a Transition at very low potential Generally speaking, at very low potentials, J (V) exhibits a transition at the threshold voltage (Vn) going from intrinsic conduction to SCL regimes (see Figure VI-tO) . In effect, at sufficiently low potentials, the concentration of injected carriers (n) remains considerably lower than intrinsic carriers (no), which are generated by thermodynamic equilibria, and conductivity follows classical the ohmic law: In = qnOllVa/d = qnOIlEa. Vn, the voltage at which the current density (J) starts to deviate from the ohmic law towards a quadratic SCL law, is such that (In)v=vn = (Js)v=vn, which yields 8 go d 2

Vn = "9

-%-. In effect , only Vn may be used to estimate no.

b Towards high tensions: transitions between regimes

a With a blocking contact The current density

JOsT at an electrode follows, primarily, the law of thermodynamic emission, while in the mass of the material , the

J

JU= q no II VJ d

I

t..-J J • =-98 E,.." VI -"d

."

\

J

V.

Figure VI· to. Transition tension from equilibrated thermodynamics to the SCL law.

144

Optoelectronics of molecules and polymers J

Resultinj;

J lkT

v. Figure VI-H. Possible transitions between different regimes with increasing potential.

curre nt is limited by space charge and tend s towards J SCL possibl e:

= J s . Th e following are

if J SCL < JOsT, the SCL law is limiting and therefore repre sent s the current through the structure; 2 if J SCL > J OsT (in V a = VTl --+ 2) the thermoelectri c emission law (Richardson) conditio ns the current goi ng thro ugh the struc ture; and 3 When E a is very high , J OsT gives way to field effect em issions and then to tunnelling emissions. If J OT repre sent s the curre nt densitie s emitted at the electrodes, depending on whether J SCL < J OT or J SCL > J OT the current through the structure will follow, respectively, the SCL law or the laws gove rning the electro des (JOT) . ~ If the contact is ohmic By principal J == J SCL (few traps) (or, as we shall see in the followi ng Section , J = h FL if there are many traps). At highly elevatedfield strengths, power laws for E (E 2 for SCL, Em for TFL) ca n res ult in extreme ly high volume curre nts with J SCL higher than the inj ection curre nts (J Electrodes) generated at the levels of contacts, which in them selves never perfe ctly ohmic as they are never actually infinite reserves of current. At extremely elevated field strengths, J SCL > J Electrodes is possible and throughout the struct ure, a transition from SCL (or TFL) to an emission law by tunnelling effects (co ntrolled by the electrode) ca n appear.

4 Insulators with traps and characteristics of trap levels a Origin and distribution of traps in organic solids Organic solids are often far from 'ideal' and contain traps due to imperfections which interact with inj ected charge carri ers . The traps control the current volume of carriers and the characteris tics of J(V). There are two types of energetic distributions of traps which are generally cited: • •

due to discreet trap levels genera lly due to chemical impurities; a con tinuous distribut ion of traps, generally due to structura l default s, of exponential Gaussian form with max imum den sity at the band edge.

VI Electron transport properties

145

These traps-neutral when empty and charged when full-exert a weak radius of action and can be spatially modelled by an exponential law. The electric field E, only slightly modifies (to the order of llET ~ meV) the potential barrier (U) as shown in Figure VI-12-a, and trapping-detrapping follows Randall and Wilkins's law, that is p = voexp(-U/kT) . We should note that two depths of trap can be considered from a thermodynamic point of view, as presented in Figure VI-12-b, with respect to electron traps. That is to say that the electron (and hole) Fermi pseudo-level (EFn and EFp) has a factor of degeneration (g, or gp) due to occupation by an electron or hole. The probability of an electron being captured by a trap at energy level Et is given by: fn(Ed = 1/[1

+ g;;-l exp{(Et -

EFn) /kT}] .

From the probability of trap occupation, we can distinguish between two types of trap: •

•

shallow traps corresponding to levels Et(=U) and situated , for electrons, well above the Fermi pseudo-level of a weak applied field (EFnO) (giving a state approaching that of a thermodynamic equilibrium such that EFnO ~ EF). In a regime close to equilibrium, most shallow traps are empty, while outside equilibrium (strong injection with EFn ---... Et), some shallow traps are filled; traps with levels E, below EFnO for electrons, in other words deep traps, are mostly filled with carriers at thermodynamic equilibrium, and in a regime of strong injection (E, < EFnO < EFn), practically all traps are filled.

b Order of size of trap densities • •

Trap density in a solid is typically of the order N, ~ 10 18 cm >' . If we take for example a cfc lattice, which contain s an average of 4 atoms per unit and has a unit parameter of ca. 0.4 nm, atomic density is ca. 41(4 x 10- 8) 3 ~ 6 x 1022 cm- 3 . The ratio d::~~td~~~~:~Y ~ 0.2 x 10- 4 i.e. less than I default every 10000 atoms .

x

Insul at or (I ) (II)

(b)

Figure VI-12. (a) spatial modelling of a neutral trap; and (b) electron injection levels in an insulator with deep and shallow electron traps.

146

Optoelectronics of molecules and polymers

c Trapping-detrapping statistics for traps in the discreet level E t The volume density of trapped electrons (n.) can be given using nt = Ntfn(Ed in which fn(Et) is the probability of electron capture by a trap of energy level Et. Note here that Et == D, the trap depth; see Figures VI-12-a and b. Additionally, if c is the capture coefficientof a trap and n is the density of mobile carriers (with n = Nc exp[-(Ec - EFn) /kTD, the number of carriers trapped by necessarily empty per unit time is equal to en. As (N, - n.) is equal to the volume density of empty traps, the number of trapped carriers per unit volumeand unit time is therefore cn(N t - n.), which represents the level of total trapping. The alternateprocess though is that of detrapping in which traps liberate p electrons per unit time (p is givenby p = Vo exp( -D / kT)). As nt is the number of occupied per unit volume, the number of electrons detrapped per unit volume and unit time is pn., which represents the overall detrapping. At equilibrium, the degree of trapping and detrapping is equal and therefore we have Nt or nt = ----:.(4) 1+ p/cn where p/cn the ratio of probabilities of detrapping from a full trap to trapping by an empty trap. We can now envisage different scenarios: •

nt « N, (low degree of trapping): we have ~~

= I+~/cn

«I i.e. p/cn

»

I,

which leads to: ~ ~ p)cn ' that is n p Vo = = -exp(-EtlkT). nt cNt cNt

-

•

Numerically, with E, = 0.25 eV, N, ~ 1018 cm- 3 , c = 10- 10 cm' s-l, Vo = 14 1010 s-l, and at T = 80 K we have .!l n, ~ 10- , while at T = 300 K we obtain .!l ~ 10- 4 . In both cases there is a very high proportion of trapped electrons with nt respectto free, mobileelectrons, even though there is a lowconcentrationof traps. nt ~ N, (high degree of trapping): this occurs when p --+ 0 (very low detrapping probability) i.e. when E. == D » kT and is due to very deep traps, or at a limit, very low temperatures.

- - - - - - - E, FigureVI-13. Location of energy levels forshallow and deep traps.

VI Electrontransport properties

147

5 Expression for current density due to one carrier type (Jsp) with traps at one discreet level (E t ) ; effective mobility a At low tensions no » n and the density of thermally generated carriers (no) is well above the density of injected charges (n) and the current undergoes runaway (current emitted or injected). In the presence of traps of density Nt. we have no = nor + nOt in which: -

-

nOf = N, exp(-[Ec - EFnO] /kT) and is the density of free charges (not trapped) of mobility !-lOf (with a Fermi pseudo-level in this low tension regime EFnO ~ EF: Fermi level at thermodynamic equilibrium). nOt = Ntfno(E t) = I Nt density of charged traps . l+gn exp[(E, - EFno)/ kT ]

Ohm's law, with respect to free carriers, is (with

hI

uor = !-In)

= q nOf !-lOf t ,

(5)

a Two limiting cases •

•

Shallow traps: E, > EFn > EFnO, so exp[(Et - EFno)/kT] »1 and nOt ~ 0: no = nOf + nOt ~ nor. Ohms law remains unchanged when there are no traps: lQ = qno uor E. Deep traps : E, :::: EFnO , and nOt i= O. For Ohm 's law, beyond the general eqn (5), we can also write

(6) which defines the effective mobility, the value of which can be obtained on equal ising eqns (5 and 6) to give !-leff = n n+01'n uor Of Ot ~

•

•

Important comments General expression for mobility: In a general manner El = nor/nor + nOt with !-leff = El!-lOf . With shallow trap s (not ~ 0 and no ~ nor): El ~ I and !-leff ~ uor . Under a continuous regime V < VQ, and ohmic conduction is predominant but does not prevent carrier injection , which is in the minority at low tensions. Accordingly, a trans itory regime is generated corresponding to trapping of injected carriers which is associated with the appearance of current peaks; it is only once carriers are trapped that linear, ohmic conduction can be observed. If we take the curve of a second characteristic leV), we can then see a much smoother transition, practically without peaks (as trap s responsible for current peaks are filled) .

b Form of J at higher tensions (injection) At higher tensions, the Fermi pseudo-level is EFn and we have : - n = N, exp( - [E, - EFn]/kT) for density of free , untrapped charges with mobility !-If = !-In , - n, = Ntfn(E t) = I Nt for density of trapped charges. l + gn exp[(Et- EFn)/ kTJ

148

Optoelectronics of molecules andpolymers

Thus e ~ en = n/tn + n.) and j..leff = en j..ln. Under a regime of charge injection, whether traps are deep or shallow, injected carriers are distributed between free and trapped carriers. The result is the mean effectivemobility,whichcan equally be obtained through using a Poissondistribution.

a Effective mobility with Pmobile = nq and Ptrap Pmobile + Ptrap

dE dx With J

= nt q, Poisson's equation gives: (n + ndq

-----"-=-----= E E

= n q v = n q j..ln E: dE dx

In+nt nj..ln E

J j..leffE

(7)

E-=---=-

with j..leff

n

= j..ln-· n+nt

(8)

As n + nt > n, in all cases, j..leff < j..ln . (When there is a low degree of trapping, it is possible to write j..leff and as ntln

»

I, j..leff

I = j..ln I + Intln = I + eNtip = p +PcN ;

«

t

j..ln).

~ Expressionfor J Integrationof eqn (7) gives,for E, a solutionwhichresemblesthat of eqn 0), with the precision that the effective mobility (j..lefr) replaces the mobility j..ln (== j..l by notation). J s (denoted in the presence of traps as Jsp) is thus given by eqn (2), in which j..l is replaced by j..leff:

J

9

sP

V2

9

E2

= gEj..leff"d3a = gEj..leW""da '

(9)

Comment Current density can also be written either as J = q n j..ln Ea, or as J = q(n + nt)j..leffEa . The equality of these two relationships gives j..leff = en j..ln again with en = n/(n + n.). By comparing eqn (9) with (9')

shows that (n + n.) ex: E, and that in (9'), J is not ohmic. Physically speaking, this is quite normal, as the concentration (n) depends upon the value of E, under a high tension regime. y Trap effects on conduction

-

-

If the concentration of traps N, increases, in accordance with eqn (4) nt also increases, such that j..leff (given by eqn (8» decreases along with Jsp; the material no longer appears insulating; IfE, increases,or ifT decreases, p drops and nt increases;once again j..leff decreases along with J sP '

VI Electron transport properties

149

& Frequency limit for Jsp as expressedfor a static system Given eqn (3), which concerns a system without traps v < vc = s = IJ. ~ a , [L must be replaced by [Leff, and the limiting condition for eqn (9) corresponds to frequencies such that:

t i

V

<

Vceff

= -

v,

I

tseff

3 [Leff = - - - 2- .

4

(10)

d

As [Leff < [L, Vceff < Vc and can be explained by the fact that the transit time increases as carriers spend more time in traps such that transit time in the presence of traps . tseff > t and thus vceff = tseff 1 . the quotrent . (10) (tseff ) IS < vc = ~1 . 0 n perforrrung m s, we obtain v ceff Vc

[Leff [L

n

= n + nt

JsP Js

Using the calculation in Section 4-c above, in a system with low level doping it is evident that Jsp ~ 3vi at 300 K and J sp ~ 1O- 14Vi at 80 K, which yield limiting frequencies of, respectively, Vceff ~ 4 MHz and Vceff ~ 7 [LHz. At low temperatures, we are limited to [LHz, which correspond to excessively long signal periods (greater than 106 s).

c Characteristics of JsP = f(V) for levels of discreet, shallow and deep traps (Figure VI-14) a System with discreet shallow traps When there is a low enough level of traps and the system is placed under a tension sufficiently high, so that no is negligible with respect to injected charge density and the Fermi pseudo-levelis below the trap energy (E) logJ

9 v2 J s =-£" 8 rn -d3

o

C'

Vn(~n)

IVll(e~ )

VTFL(Nt) I

IVTFL(Nt)

log V I

FigureVI-14. SCL law inpresence ofdiscreet, shallow traps.Atlow tensions, thelaw isohmic (In) ; at higher tensions (V > Vn) current follows Jsp law, which is the SCL law modified by carrier mobility which, on average, reduces i.e. !-Leff = !-Lnen with en < I. When all traps are filled (V = VTFL) thesystem returns to following the SCL law (current Js).

150

Optoelectronics of molecules and polymers

level, the above cited SCL law (Jsed tends toward s a new law (Jsp) which can be written n 9 V2 en = - - : Jsp = -SE(..lnen3 n+nt d and varies as V 2 /d 3 but with a reduced charge carrier mobility (Figure VI-14) . At even higher tensions (V 2: VTFd all traps fill and the system returns to the normal SCL law. Transitional tensions, VQ and VTFL , are shifted towards higher tensions when trap density rises (N; > N, and e~ < en). For very shallow traps, where E, » EFnO, and at thermodynamic equilibrium the concentration of carriers in traps levels is given by: ntO

=

N

t ~ O. 1+ g;; exp[(E t - EFno) /kT] 1

And the density of non-occupied traps at thermodynamic equilibrium is thus N, ntO ~ Nt; application of a tension up to V = VTFL results in the filling of initially empty N, traps . We can use Poisson 's equation to detail the tension VTFL in a corresponding field ETFL : d~~ = q~l . By twice integrating this equation from 0 to d, we arrive at VTFL =

J~ ETFLdx = q~~d2 . (From VTFL, it is possible to derive Nd.

~ System with discreet, deep traps The mechanism for a system with deep traps shown in Figure VI-IS exhibits, at not too elevated tension s, very low effective mobility and thus a low current limited by space charge effects (essentially associated with trapped charges) . The ohmic regime continues up to the point at which all traps are filled, which occurs at a sufficiently high tension (VTFd , and then current follows the SCL law, characterised by the current density Js . For deep traps , we have Et < EFnO , and at thermodynamic equilibrium the concentration in trap levels of carriers is

ntO =

Nt I

I + g;; exp[(E t - EFno) /kT]

i=- O.

logJ

_--t--IJ VTt'I.

lI

=q " or JlorV/d

I

logV

Figure VI-IS. Current tension characteristics for system with discreet, deep traps.

VI Electron transport properties

151

At thermodynamic equilibrium, the density of unoccupied traps is thus N, - nOt . Withapplied tension VTFL resultsin the fillingof the (N, - nod traps and an associated charge density of q(Nt - nod. Poisson's equation allows determination of

=

VTFL

l

d

o

ETFLdx

q(Nt - nto)d2 28

= -"------

y Comment We should note that while these characteristics have been observed, especially for systems with shallowtraps, only the firsttwo parts of the curve generally appear (region in OABC of Figure VI-14); at higher tensions, diodes have a tendency to undergo premature degradation!

6 Deep level traps distributed according to Gaussian or exponential laws a Distribution form

We can consider different forms of distribution for traps, that of exponential or Gaussian. Here we shall develop the former, and in the next Section, the latter. We can suppose that the density of traps per unit energy,centred about energy E, is of the form: geE) = (Nt/kTt) exp(- [E, - E]/kTd in which N, is the total density of trap levels and T, is a constant characteristic of the decrease in trapping energy with depth E. Taking Tt = Et/k = mT in which E, represents the characteristic energy of traps with respect to Ec and m, also, is characteristic of trap distribution, given that for a high value of m, the decrease in distribution is less than that of Boltzmann, in which m = I. When T, » T(E t » kT and traps levels are mostly under E, giving rise to a nondegenerate distribution of traps represented by Fermi-Dirac 's distribution), we can suppose that for -00 < E < EFn we have f, (E) ~ I and for E > EFn we havefn(E) ~ o(as if T = 0). Given these hypotheses, the concentration of filled trap levels is E max

nt =

1

g(E)fn(E)dE

E min

=f

EFn - 00 .

With n = N, exp(-[Ee

-

-Nt

kT,

e - E) e Eexp(- dE = Nt exp ( - E - EFn) . kT, kT,

EFnJ /kT) we obtain

_ (n- )T/TI_-Nt(-n)l/m

nt-Nt

Ne

Ne

...---

EFn

kmp';tra~s

.-----J1F-_. __ ._.------

( +F u ll traps Figure VI-16. Representation of an exponential distribution of traps.

152

Optoelectronics of molecules and polymers

b Form of saturation current for an exponential distribution (J exp ) On multiplying the two parts of the Poisson equation by [(m obtain:

(

+ 1)/mHE(x)]l /rn we

dE(x) d[E(x)](rn+1) /rn m+ I q m + I) - - [E(x)]l /rn_ _ = = _ _ [E(x)]l /rn_(n m dx dx m E

+ n.) .

If nt » n (that is N, » Nc) and E(O) = 0 and n = Jexp /qJ.1nE(x) (denoting the current density in this exponential distribution of traps by Jexp), integration over the interval [0, h] gives us: E(h) =

[(m + I) m

qN t E

(~) l/m] m"!-l qJ.1nNc

On using the confined integral (which gives limiting conditions) Y = arrive at: J

fad E(h)dh we

rn 1 E )rn y + 2m-+-l)rn+1 ( -m- - (I-rn ) N exp - q J.1n c ( m + I m + I N, d2rn+1.

The resulting equation below is generally known as the Tapped Charge Limited Current (TCL) law and its shape is shown in Figure YI-l7. JTCL

= Jexp ex Y rn+ l / d2rn+ 1.

Qualitatively speaking, with increasing current the Fermi pseudo-level increases towards Ec (bottom of conduction band) and traps below EFn fill up. There is a progressive reduction in the number of empty trap s, and fresh injected charges are increasing distributed towards free states (in which charges have density n). The effective mobility, as great as n is high, increases with Y following a high power law

logJ

(E

Figure VI-17. Current tension characteristics for exponentially distributed traps.

VI Electron transport properties

153

V m+1 in which m, as we alread y have seen, is characteristic of the mean depth (E) of trap levels Et = kT t = kmT . When all traps are filled, a transition from a TCL current (with carriers exponentially trapped) to a SCL current (with non-trapped carriers) takes place. The tension of this transition (VTFL) can be obtained at 1s = h CL, that is 1

2 _qd- [ -9 -N (m+ VTFL - - l) m(m+1 - -- )l+ l] m-l E 8 N, m 2m + I

:n

We should note therefore that the two slopes of In 1 = f(ln V) for J, and h CL are different in contrast to those of 1s and 1sp which give parallel lines of slope 2. As we have done for traps associated with discreet levels, we can now determine Vn, the tension characteristic of a transition from the ohmic regime at low tensions (following 1n = qnOff.ln to a TCL regime. Thus Vn can be obtained when 1n = 1exp, that is:

¥)

v« =

2N

qd E

I t

(nOf)ffi N,

m+l

(~) (~)fi1 m

2m + I

c Example of exponential distribution The example of copper o-phtalocyanine is used [Kao 81]. Empirically obta ined results, detailed in Figure VI-1 8, showed that: •

in the non-ohmic zone , the slope (m) decreased with increasing temperature s (as mT = T t = Etlk = constant for a given material and as T increases, m decreases);

urrent CAl

lO- s

0.01

0.1

Figure VI-IS. Idealised representation of Jexp

10

= f(V) for ex-CuPe.

154 Optoelectronics of molecules and polymers • •

Y Q increases with increasing T; Nt ~ 10 15 cm- 3 .

d Gaussian trap distribution A Gaussian trap distribution can be denoted by

in which E tm is the level at which there is the highest trap den sity and at standard deviation. Two types of trap distributions can also be considered:

• •

Shallow traps which are supposed to be such that E tm > EFn and any current resembles Jsp (due to discreet trap level) i.e. JG ex y 2/ d 3 . Deep traps correspond to E tm < EFn and the current resembles that of J exp , and the parameter m should be replaced by a parameter of the form: m ---+ (I + 2Jt~ / 16k 2T2) 1/ 2 [Kao 81] which the expression for current remains of the form JGaus. ex ym +l /d 2m+ 1.

V Double injection and volume controlled current: mechanism C in Figure VI·2 1 Introduction: differences in properties of organic and inorganic solids Here we shall review only media in which there are only direct band to band recombinations, termed 'bimolecular' [Kao 81] but no trap s or recombination centres. The level of band to band recombination (R) is defined by the equation ~~ = ~ = -R = -CRnp where CR = (va R) and is the recombination con stant , aR is the effecti ve recombination surface (the sectional area through which a carrier must pass to recombine ) and v is electron and hole velocities. If carriers hop over barriers more by electric field effects than thermal agitation then we should denote their velocity, which depends on the applied field and carrier mobility, using v (ILn + ILp)Ea.

=

a Inorganic solids The carrier mobilities are large and con stants CR are low. In fact, the latter is generally so low that bimolecular recombinations ca n be con sidered negligible, and in order to favour recombinations-which are no longer 'bimolecular' --centres must be intro duced into the material. The velocity (v) is of the order of thermal electron and hole velocities and is independent of the applied field. In addition, for a volume controlled current (YCe), the negative and positive space charg es at the electrodes are of the same order of size as those under SCL for simple charge injection with overlapping

VI Electron transport properties

155

( a)

Low OR: 'injected pia rna' Figure VI-19. Double injection and spatial distribution of charge densities in: (a) inorganic and; (b) organic materials. in the crystal volume (without recombining), and therefore mostly cancel each other out (' injected plasma' in Figure VI- 19-a). The result is a current which is no longe r limited by space charges and can thus reach high values.

b Organic solids In contrast to inorganics, carriers in orga nic solids exhibit low mobilities and high values of the constant CR, so 'high density injected plasmas' become highly improb able. Due to the high value of OR genera lly enco untered , there is no zone in whic h electrons and holes can overlap (if there were, there would be an infinite recombination current). The result is that electron current exists only at the cathode and hole current only at the anode: the two meet and annihilate eac h other at a plane separating the two region s, as shown in Figure VI-19-b.

2 Fundamental equations for planar double injection (two carrier types) when both currents are limited by space charge: form of resulting current J vcc (no trap nor recombination centres) a As for SCL, the following hypotheses are used (as both carrier currents are limited by space charges): -

-

the band model is applicable; E, is sufficiently high to consider J intrinsic and J Oiffusion negligibl e; mobil ities IL and E are not modified by Ea ; and there is perfect ohmic contact (infinite charge reserve) at the metallic cathode (at x = 0) and anode (where x = d). We therefore have: nCO) = 00; p(d) = 00; E(x

= 0) = E(x = d) = 0 ; f~ E(x) dx = - V.

b Parmenter and Ruppel 's equation: general example In general, doubl e injectio n is assumed to be governed by: -

equations for current: I n

= n q ILn E, Jp = P q ILp E and J = Jo + Jp ;

156

-

Optoelectronics of molecules and polymers

I dJn I dJ equations of continuity: - = Rand - - ---.!: = R; and q dx q dx dE

p

q

Poisson 's equation: - = - = -[p(x) - n(x)]. dx E E Given the preceding limits and some rather long equations detailed elsewhere

-

[Kao 81, p 259], the resulting current (Jvcc) is given by Jvcc = ~E lteff ~ in which Iteff is the effective mobility which is of the complicated form Iteff = (8q Itn Itp/9E( VoR})f(a ,~ , Itn, Itp), with a = 2q/E, ~ = (VOR). c Limiting case 1: {V<JR} is low (as for inorganics) When OR is low, Iteff takes on a high value: •

•

poor recombination is not an obstacle to the penetration of electrons and holes within the sample, and the condition of overall neutrality is retained within the volume given for n ~ p ('injected plasma' shown in Figure VI-19-a); and there is no space charge to limit current so Iteff and Jvcc are high.

d Limiting case 2: CR and {V<JR} are high (<JR high, as for organic solids) Here we have two SCL currents with a weak overlap of space charge s. Jvcc can be calculated directly when CR is large [Hel 67] and recombinations are highly efficient and, consequently, there is a very thin space charge overlapping layer. We can thus consider that for the 2 SLC currents produced at each electrode, the two converging currents annihilate each other. Sequentially, it is assumed that: •

tensions and the thickne ss of layers for each-well separated-eharge type can be summed: L = Ln + Lp et V = Vn + Vp; (1)

•

the electric field does not undergo any great discontinuity in going from one part to another in the recombination zone and therefore (2)

•

current associated with electrons (over Ln), or with holes (over Lp), is such that I n = J p = Jvcc = constant. With each J, being oftheform Ji = (9/8)Elti Vf /Lf: I

r, r,

=- =

Itn(V n/Ln)2(1 /L n) 2 Itp(V p/Lp) (1/Lp)

Itn t.,

= --, Itp L n

so Itn/ltp

= Ln/Lp = Vn/Vp.

(3)

It should be noted that when Itn > Itp then Ln > Lp and the more mobile carriers travel the furthest. From eqn (3),

v, = Vpltn/ltp , which inserted in eqn (1) gives Vp = V/ (1 + Itn/ltp).

(4)

VI Electrontransport properties

157

Placing Vp in Jp = JyCC finally gives JyCC = ~£(I-ln + I-lp) ~: . The resulting current is the sum of two individual currents limited by space charge, and therefore the effective mobility is actually the sum I-leff = I-ln + I-lp ' As shown in Figure VI-19-b, the electron current exists only at the cathode and the hole current only at the anode; on meeting they annihilate one another at a plane which divides the space and has a position determined by the values of I-ln and I-lp . We can finally note that, like I-leff, JyCC is relatively weak .

Comment if I-ln » I-lp or I-ln « I-lp we have, respectively, either I-leff ~ I-tn or I-leff ~ I-lp and again find that current is limited by space charge (SCL) associated with a single carrier type, as is often the case in organic solids . However, we should note that often in these media, I n i= J p and the preceding calculation is therefore not necessarily appropriate (even though-as we shall see in later Sections-optimisation of the efficiency of OLEDs can result in I n ~ J p). e Limiting case 3: real example of continuously high but constrained CR CR does not tend towards infinity, imposing a degree of overlap between the two space charges [HeI67]. n(x) can therefore be expressed as a function ofW, the size of the recombination zone, for which:

• •

J = I n + J p = q[n(x)l-ln

+ p(x)l-lp]E(x) =

constant; and

(5)

taking into account that the recombination zone is actually rather thin with respect to the overall film thickness, the electric field can be assumed constant, so the continuity equation gives, for electrons: charge -q and velocity Vn = -l-lnE,with dn dp . ld s: · . m . dt J n = -qn Vn = qnl-lnE . B nngmg = dt = - R = - C R np Yle (dJnjdx) = ql-lnE(dn jdx) = qCRn p.

(6)

(Similarly for holes, (dJp/dx) = ql-lpE(dp /dx) = -qCRn p). Using eqn (5) to eliminate p in eqn (6) derives n(x) : n(x) =

n oo 2q E21-lnl-lp with W = , or as 1 + exp[2(Lh - x)/W] JCR

E = (3 j2)V / L, W =

4ql-lnl-lpL CR£(I-ln + I-lp)

in which, by definition, W is the size of the recombination zone . The size of W is inversely proportional to CR, (as in the limiting case 2 shown detailed in Section d) . 3 Applications a Practical application of double injection in organic solids From the relationship for current density in organic solids for double injections with a we can see that if I-ln » I-lp or I-ln « I-lp , high value of CR, JyCC = ~ £(I-l,n + I-lp)

D'

158

Optoelectronics of molecules and polymers

we obtain I-leff ~ I-ln or I-leff ~ I-lp and current is limited by space charges (SCL) associated with single carriers . This is what happens in organic solids, where with two SCL currents we have either : •

•

at I-ln » IL p (as in molecular solids such as Alq3), I-leff ~ I-ln and the recombination zone is situated close to the hole 'injecting' anode while the current is essentially an 'SCL' current of electron s; or at I-ln being around two orders lower than I-lp (as in conjugated polymers in which poor electron mobility is generally associated with traps due to oxygen impurities [Gre 95, p74]), I-leff ~ I-lp and the recombination zone is situated close to the electron 'injecting' cathode while the current is essentially an 'SCL' current of holes.

b Both electron and hole currents are limited by electrodes (rather than SCL effects as above) The problem can be treated as a superposition of two individual currents, although modification of the level of injection can occur at the contacts if electrons-hole recombinations are incomplete.

c Application to electroluminescence To ,?btain electro luminescence it is essential to .control the level of injection and transport of the two currents , each associated with a different carrier, so that: -

-

the currents are at equilibrium so that there is no single dominating current which could otherwise traverse the diode without meeting current of the opposite sign; and the currents meet each other in the volume of the material and, if possible , not near the electrodes where recombinations do not yield radiation . As we shall see, various strategies have been devised :

-

-

if current is electrode governed, barriers at the electrodes can be adjusted for electron s (Ca) and holes (ITO) so that currents are of the same order of size. Barriers can be brought to the same low level by adjusting either the work function of the chosen metal electrode, the electron affinity or the ionisation energy of the organic volume; and if current is controlled by the volume of the diode, then we can choose layers of materials which go in between each electrode and the volume of the material which exhibit mobilities equilibrating the two SCL currents. Recombinations can then occur within a layer specifically optimised for high quantum emission yields.

Another strategy consists in generating a confinement layer which can be formed from 'heterojunctions' which exhibit at least bilayer structures allowing recombination of opposing carriers to be confined at a plane normal to their trajectories.

VI Electron transport properties

159

T = 87 0 C Current (A) 10-6

T =34 0 C

10-8

10-10 L . . - ' - 1

--'--

2.5

----'-

---+

Voltage (kV)

5

Figure VI-20. Single to double injection transition using example of anthracene with gold electrodes (after Hwang and Kao (1973) [Kao 81D.

d Transition from single to double injection At low field strengths, one electrode can already inject for example electrons, while the other cannot yet inject holes . At a higher value of E, (and thu s from a single field) the second electrode can in turn inject (as shown in the example in Figure YI-20below for anthracene with gold electrodes): • •

at tensions Y < VTH only for electroluminescence, J ex: v- , a characteri stic of a solid with single (hole) injection (and discreet hole traps) ; at tensions Y > VTH, J ex: V'' where n > 6, a characteristic of a high current which can be asso ciated with electron injection by field effects, which in turn can result from a sufficient accumulation of space charge at the cathode to favour electron emi ssion by tunnelling effects.

When there is double injection , the currents are summed, but if one is con siderably greater than the other, the largest will determine the resulting law.

VI The particular case of conduction by the Poole-Frenkel effect Solids which contain ion isable centres, which act as though they are Coulombic trap s, can exhibit conduction through the Poole-Frenkel effect.

160 Optoelectronics of molecules and polymers

1 Coulombic traps When these Coulombic trap s are empty they are positi vely charged (having charge + q prior to capturing an electron) and neutral when full. Typically, the potential energ y of an electron (located by x) in a Coulombic trap (centred at xo) ca n be expressed as: V ex) ~ - q2/4ml x - xol. If E, repre sent s the bonding level, the ionisation energy (of modulus equal to Ed is modified (by a drop of ~EpF ) by an applied field E. Given that V = - qE(xo)(x - xo) - q2/4 m lx - xol, applying the condition 0 = (dV/ dx)x=xM allow s determination of (x - XO)M = [q/4JtEE(xo)] 112 and ~EpF = (q3 E/ m) 1/ 2 = ~PFEl /2 , with ~PF = (q3/ JtE) 1/2.

The initial barrier (Ej) is now in the applied field (E) EI - ~EpF. For organic materials (insulators), Eis low ( ~ 3) and (x - XO)M is 'high' : (x - XO)M (in 11m) ~ 40 / JE (E in Y m- I ) that is (x - XO)M ~ 100 nm . In addition, ~EpF is also high , and ~EpF (in eY ) ~ 8 x 10- 5 JE. With E in Y m- I , ~EpF ~ O.leYforE ~ 105Ym - l . If E increases, (x - XO)M decreases and the electronic emi ssion process tend s toward s tunnelling effects.

2 Conduction due to Poole-Frenkel effect (as opposed to Schottky effect) The Poole-Frenkel effect, sometimes called the Schottky internal effect, occurs within solids where electrons move from ioni sable centres. The ions, bein g near immobile, do not contribute directly to the conduction. If nCO) is the conc entration of electron s trapped within a solid in the absence of an electri c field, once a field is applied the barrier ob served by electrons is reduced by an amount ~ EpF = ~PFEI /2 , with ~PF = (q3/m) 1/ 2. The concentration ofelectrons thu s

Figure VI-20. Electric field effect on a Coulombic trap.

VI Electron transport properties

--

161

-q2

,·.....- ....... 1

= -4rrex -

q2

s<...

t>, = -16rrex

In J Schottky slope:

m~

Poole-Fr enk el slope:

£ /7~ r-

...,

Space cha rge

~~.!:L kT lt E

cITcct

Figure VI·2l. (a) Poole-Frenkel and Schottky potentials; and (b) representation comparing Poole-Frenkel and Schottky laws.

changes to neE) = nCO) exp(ilEpF /kT) = nCO) exp(~PFEI /2 / k'T), and conductivity then follows an exponential law, exp(~PFEl /2 / k'T). The Poole-Frenkel law is of the same form as that found for a Schottky emission. However, the latter law follows more exactly exp[(~PFEI /2 / 2kT], as in place of a Coulombic force (l /4mx 2), the image force (l /4m[2x]2 = 1/16mx 2) intervenes in Schottky emissions (Figure VI-21-a). In both cases a linear law is retained, that is InJ = f(JE) (in Figure VI-21-b), and in reality it is difficult to distinguish between the two mechanisms even though the slopes differ by a factor of 2 and the physical origins of the two laws are different (the Schottky effect is a limitation of current by electrode emission, while the other is a current limitation due to the material volume) . This inexactitude (in terms of the one-dimensional representation of transport) is due to incertitude in the value of s (for the field under study) , arising from the possible stepped placement of wells giving a detrapping on a neighbouring site rather than in the conduction band . As we shall see in Chapter X, the Poole-Frenkel law is in fact particularly useful in justifying transport laws proposed for organic LEDs, where mobility under an applied electric field is thought to follow a law of this type .

VII Optical processes in molecular and macromolecular solids

I Introduction In this Chapter, we shall look at the origins of optical properties of: •

•

a solid matrix into which has been inserted transition metal or rare earth atoms. While this may not directly concern organic solids, it is a field of considerable importance in optoelectronics in general, and has given rise to applications for example, in telecommunication systems and electroluminescence; and molecular or macromolecular solids .

As an aide-memoire, Appendix A-8 details the successive appearance of energy levels within an isolated atom following perturbations of decreasing intensity. Optical propertie s, in general, are governed by: • •

the separation of energy into quantum levels in atomic or molecular based structures which condition the wavelengths of emitted or absorbed light ; and the transition rules, otherwise termed 'selection rules', which determine permitted transitions between quantum levels.

The above rules are based on the conservation of total kinetic momentum within a system associated with the emis sion or absorption of a photon when an electron transfers from one energy state to another. However, in a more rigorous manner, we can also say that any variation in the kinetic momentum of an electron implicated in a radiating transition should equal I . By definition, any radiating transition is one which employs the emission or absorption of a photon, which is a boson type particle . Indiscernible particles do not follow the Pauli principle [Bla 67, p8] and any number of bosons can occupy the same quantum state, however, the state function must remain symmetric-with respect to permutation of co-ordinates between two particles-and associated with a whole spin number (from the symmetry postulate [Coh 73, p1374] and thus also Pauli's principle [Lev 68, p99]). So, for a photon, S = I.

164

Optoelectronics of molecules and polymers

incident photon with circular polarisation

atom following absorption of incident photon

Figure VII-I. Schematised atom-photon interaction in which circularly polarised photon transfers its kinetic momentum to an electron at the periphery of the atom (process of absorption) . Variation in kinetic momentum of the electron accords with ~I = ± I.

Selection rules governing radiating transitions take into account gains or losses in kinetic momentum due to photon absorption or emission, respectively. The process is schematised in Figure VII-I .

II Matrix effects due to insertion of atoms with incomplete internal electronic levels (transition metals and rare earths) 1 Electronic configuration of transition elements and rare earths Mendeleev's periodic table generally follows the sequential filling of atomic levels by electrons as given by Klechkowski's rule and detailed in Appendix A-8 [Am 91, Chapter 9]. Columns IA and IIA correspond to the s block, while columns IlIA to VIlA , along with column VIII for rare gases , correspond to the p block. Horizontal levels in the table are filled with respect to the number of electrons added , respectively, to sand players. The d block, however, corresponds to transition metals which follow the series nd (n = 3,4,5 and 1 = 2) filled as (n + l)s. If we take the example of scandium, for which Z = 21, and using (Ar) to denote the base electronic configuration of argon, we can write its configuration as [Ar] 3d I 4s 2 . The result is that for these 'transitional' elements, the electronic layer nd below the outermost layer is incomplete. Rare earths, which belong to block f, exhibit a similar electronic disposition. Their intemal4f layer, which is at [n + 1] = 7, is incomplete and is filled only when the series 5s, 5p and 6s reach the full configuration of 5s 2 , 5p6 and 6s2 . An example is that of erbium (Z = 68) which has the configuration [Xe]4f I26s2 . Comment: The use of Klechkowski's rule does involve some exceptions, which incidentally do not detach notably from its use [Am 91, p75]. For example, chromium (Cr) and copper (Cu) have configurations that stray slightly from the rule each having only 1 4s electron instead of 2 and an additional 3d electron.

VII Optical processes in molecularand macromolecular solids

165

2 Incorporation of transition metals and rare earths into dielectric or a semiconductor matrix: effects on energy levels The optical properties of a matrix are controlled by the conventional band scheme, drawn in reciprocal space and shown in Figure VII-2 where E = E(k) . The size ofthe forbidden band generally controls emission and absorption spectra . Vertical (direct) transitions are the most probable, when possible, as given the same k position at the extreme edge s of valence and conduction bands a two particle electron-photon transition which conserves energy and quantity of movement is allowed. Low concentrations of transition metal and rare earth ions incorporated into solid matrices (for example, by ion implantation) can give rise to particular optical properties. For examples, lasing effects are observed on incorporating Cr3+ in Al203 and doping of silicon fibres with erbium can permit integration of 1.541-lm lasers into optical fibres for telecommunications. These changes in optical properties are due to the increase in degeneration of energy levels associated with the electronic configurations of transition metal and rare earth ions . In order to understand these systems, the strengths of the different effects which are brought to bear, such as exact electronic repulsions given by the Hamiltonian Hee , spin-orbit coupling given by Ho .s , and the crystal field effect resulting from the environment and characterised by Hcc, will all depend of the type of ions used.

a Rare earth ions Electrons in the 4f layer are well buried within the atom . The full, external layers, 5s2, 5p6 and 6s2, which have wavefunctions with greater external radii than 4f remarkably well protect the exterior layer (crystalline) of the 4f electrons, so in general for rare earths Hee > Ho .s. > Hcc. Up to and including Hs .o., we have more or less the same

Vertical

transition VB

~ -lL-. (a)

v

"-

Figure VII-2. Transitions in a solid with conduction band (CB) and valenceband (VB):

(a) vertical (direct) transition, most probableas is associated with only two particles, electron and photon (with kphoton = 21£ / }.. ~ 10- 3A- 1 ~ 0). The photon assures energy conservation during transition betweenVB and CB with t.E = ECrnin - EVrnax = hv; (b) oblique (indirect) transition, of little probability as associated with three particles, electron, photon and phonon (with t.kphonon sufficiently high to allow conservationof quantity of movement).

166 Optoelectronics of molecules and polymers behaviour for atoms in the isolated state as in a matrix, so the crystalline field effect can be considered negligible. Spectroscopic and luminescence properties associated with 4f internal transitions are thus little influenced by exterior perturbations, to the point where they are practically independent to the host material and even temperature. Insertion of these atoms in a matrix can be accomplished by ion implantation, although this method does necessitate annealing the material to activate the optical properties of the implanted ions. Annealing induces the formation of 'packets' of emitters which have dimensions of the order of several microns . The observed luminescence does not seem to be due only to implanted erbium ions, for example , but also their association with other elements, in particular oxygen [Fav 93]. An interpretation is given in Section 3-b.

b Transition metal ions d Layer electrons of transition metal ions are only poorly shielded from the exterior, such that perturbation due to crystalline field effects is greater than that of spin-orbit coupling.Typically, Hee > Hcc > Ho .s., and the crystal field effect can be sufficiently strong to directly interact on LS terms (degenerated [2L + 1][2S + 1] times) due to the correlation (exact interaction) between electrons in the same atom . The crystal field effect (often denoted by a quadratic potential V : V = Ax 2 + By2 + Cz 2 ) starts (before spin-orbit coupling effects) to partially raise degeneration of LS terms . In a typical case, it raises orbital degeneration leaving only spin degeneration or order [2S + 1] [Mol 81]. The obtained Hamiltonian can therefore only be expressed as a function of spin order and is denoted the spin Hamiltonian. At this level, two different cases can be envisaged: •

either perturbation introduced by an exterior magnetic field is smaller than that resulting from spin-orbit coupling, as in Figure VII-3, in which case spin-orbit coupling-which interacts after the crystal field effect-does not raise remaining spin degeneration. The resulting perturbation does not even displace levels associated with correlations, but only modifies the states (wavefunctions) corresponding to the latter. Only the intervention of perturbation by an exterior magnetic field

'--_ _ Initial configuration )'(n,l)

~y-----~I_,-)-
Correlations ofLS terms and levels degenerated [2L+l][2S+1] times.

Application of crystal field [2S+1]; rise in orbital degeneration and order of spin [2S + I]

Modification of states due to orbital spin coupling.

Increa se in spin degeneration due to external magnetic field.

Figure VII-3. Increase in degeneration when magnetic field effect is relatively weak.

VII Optical processes in molecular and macromolecular solids

167

(which introduces a perturbation Hamiltonian of the form ltog~s .iI where g, while written as a scalar corresponds to a tensor and ~ is Bohr 's magneton) raises the degeneration of energy levels; or perturbation introduced by an external magnetic field iI is greater than that produced by spin-orbit coupling; the latter simply displaces levels appearing due to the magnetic field. An example, with iron atoms , is shown in Figure VII-4 .

iI •

The structure which appears on application of H (Zeeman effect applied to solids) can be evidenced by electronic paramagnetic resonance (EPR). A study of transition probabilities between Zeeman levels yields the selection rule ~Ms = ± I [Mol 81].

3 Transitions studiedfor atoms with incomplete layersinserted in a matrix a Transition metal ions When isolated, iron has the configuration [Ar] 3d 6 4s 2 . Excited states have configurations 3d? 4s I then 3d 6 4s I 4p l and then 3d 8 . The lowest energy levels of these configurations are, respectively, 0.85 eV, 2.4 eV and 4.07 eV above the lowest , fundamental level, which has configuration 3d 6 4s 2 . Once introduced into a matrix as impurities, that is to say at very low concentrations , these elements loose one or more electrons. Generally, iron becomes the cation Fe3+ which has the unexcited configuration [Ar] 3d 5 and, 23 eV higher, has the excited configuration [Ar] 3d 4 4s I . Optical properties, which arise from differences in energy generally less than 3 or 4 eV, can only arise from internal transitions to the non-excited configuration [Ar] 3d 5 , shown in Figures VII-3 and VII-4 . Under the influence of crystal field effects , which modify energy separation s, transitions

r:y(;,i)" ~~figu~atio~ l I

I

internal transition

, '- - - - - - - - - - T '

j I! I!

~---__;:

Initial configuration "(n, I)

I

t1E= llog~Ht1Ms

i Fine Structure(FS) j

Corre lation s ofLS terms and level s degenerated [2L+ I][2S+ 11 times .

Applic ation of crystal field [2S+ Ij ; rise in orbital degeneration and order of spin [2S+ I].

Exterior magnetic filed applied raises degeneration of spin by [2S + I l.

Displacement of

i precedent levels by

!orbital spin coupling ~

withoutincreasing

! degeneration i(without considering ! hyperfine structure

Idue to nuclearspin).

FigureVII·4. Increase in degeneration for example of iron with perturbation due to magnetic field greaterthan orbitalspin coupling.

168 Optoelectronics of molecules and polymers internal to the nd configuration, as shown in Figure VII-4, can give rise to quite large separations and spectral lines. In diatomic molecules, which are the classic examples, internal electronic transitions modify interaction forces and therefore the distance between the two atomic nuclei . In other words, the passage of an electron to an excited state reduces the bonding state, allows increased nuclear repulsions and an increased distance between the nuclei. Similarly, transition metals implanted into a matrix have electronic states, characterised by their configuration, and subsequent effects such as those of crystal field, based on the position and movement of their respective nuclei at co-ordinate (r), the latter being located with respect to the mean position of a stable state at the minimum of the configuration curve E = E(r), which has a form corresponding to that of an harmonic oscillator. Energy levels within the oscillator are tied to the vibrational states of the system, characterised by quasi-particles called phonons, and coupling between the electronic states and the positions of the nuclei (i.e. the relation between the states of electrons and their ions) is called electron-phonon coupling. With increasing separation in excited and fundamental configuration curves b.r(=rexc - rr), there is a proportional increase in the electron-phonon coupling. The vertical transition, associated with the Franck-Condon principal which states that the electronic transition is sufficiently rapid to leave the co-ordination (r) for the transition metal ion nuclei unchanged, occurs between two states with the same difference in vibrationallevels (large variation in vibrational quantum number (v) with quantification of phonons) as excited and fundamental electronic configuration curves are shifted (see Figure VII-5). As shown in Figure VII-6 (transitions F-E), once an electron has been excited, it can loose its energy [Pan 71, pI67] :

E

Fundamental configuration rr

Very weak coupling

Quite strong coupling

r exe

r

Very strong coupling

Figure VII-5. Configuration curvesof different bondsfor fundamental and excitedstates.

VII Optical processes in molecularand macromolecular solids

E

169

Excited confi~uration

T

Hj

~ndamental

configuration

r

i

(a) radiative relaxation

(b) non-radiative relaxation

Figure VII-6. (a) Radiativeand (b) non-radiative relaxations.

•

•

by radiation (Figure VII-6-a). The excited system at point E has vibrational level> I and on going from E to T looses quanta of vibrational energy to the lattice in the form of phonons. Thus , thermally energetic electrons are in the lowest part of the potential curve at the fundamental vibrational level (v = I) of the excited configuration. Electrons then relax through the radiative vertical transition TH, which involves the emission of a photon with an energy smaller (and therefore greater wavelength) than that associated with the vertical transition FE. This increase in wavelength is called 'Stoke's shift' . Finally, by phonon emi ssions, the system relaxes from H to F where the electron return s to its lowest electronic and vibrational level ; or by a non-radiative pathway (Figure VII-6-b) . Here the potential curve associated with the fundamental state intercepts between points E and T the curve associated with the excited electronic state at point U. T is the fundamental vibrational level of the excited system. The system excited to E now relaxes by emitting phonons, first going from E to U through the lattice associated with the excited electronic configuration, and second from U to F into the fundamental configuration system .

b Rare earth ions As we have seen already, the interior, incomplete 4f electron layers, which according to Kleschkowski 's rule are filled last, of rare earths are protected by external, full layers, notably 5s 2 and 5p6. The upshot of this configuration is that properties of luminescence associated with internal 4f transitions, are relatively insensitive to external effects such as crystal potential and temperature from a surrounding matrix . Indeed, the configuration curves E = fer) do not to any practical extent shift between fundamental and excited states for transitions occurring between electrons of the same

170

Optoelectronics of moleculesand polymers

4f configuration. Differences between energy level s can thus be reduced to simple differences between flat levels associated with the minima of configuration curves, as detailed in Figure VII-7. Erbium is of particular interest as the wavelength of its emitted light (1.536I-Lm) is very close to the maximum transmission wavelength in silicon optical fibres (l .55I-Lm) . The emission is in fact generated by the cation Er3+ which has configuration 4f 11 [Mom 95, p35] . The fundamental spectral term is Smax = 3 x ~ = 3/2, (~ml)max = (Mdmax = 3 + 2 + 1 = 6 and so Lmax = 6. From J = L + S (as in the second half layer) we have J = 15/2. The fundamental term is thus 4115/2 while the first excited LS term following spin-orbit coupling (see Appendix A-8, Figure 2) corresponds to J = L + S - 1 = 13/2 and is denoted 4113/2. For the transition 4113/2 ~ 4115/2 (which should give a fine spectral line as it is associated with spin-orbit coupling terms), the selection rules (~S = 0, ~L = ±l or 0, ~J = ±l or 0) are well respected but the supplementary condition, ~l = ±l in which 1 is the kinetic momentum of a 4f electron undergoing the transition, is not verified (as this electron does not change configuration and remains 4f) . This forbidden transition corresponds to Laporte's rules which, as we shall see in more detail for molecules, indicates that electron dipolar transitions (through oscillating dipole radiation) are forbidden within the same configuration, as during an interior transition there is no inversion of orbital sign and the dipole moment of the transition is cancelled out because initial and final states have the same, retained parities. However, the rule is often partially violated, as mixtures of excited orbitals (associated with alternate configurations) with inverse parities are formed . It should be noted also that the probabilities of transitions between state s 4115/2 (generally denoted state T j) and 4113/2 (state T2) are in fact low and the resulting spectral lines are weak . In addition, the lifetimes of excited states (T2) are long , as return to the fundamental state (Tl) has a low probability, which leaves open the possibility of a secondary excitation from the T2 state to an even higher energy state T3. The wavelength A2 associated with a return to the fundamental state (transition from T3 to T I) is consequently shorter

V

~2(0 -Jl------~-

~:::::::::-----------(a)

(b)

Figure VII-7. For rare earth ions inserted into a matrix, we see a reduction to plateau of configuration curves (b) associated with differentexcited levels(a).

VII Optical processes in molecular and macromolecular solids

171

(E], AI)

T~

(E2, A2)

(E], 1. 1)1

TI -

-

-

-

-

Figure VII-S. 'Up conversion' using long lived T2 level.

(as of higher energy E2) than A.] associated with the excitation. Figure VII-8 details further this so-called 'up conversion'.

III Classic optical applications using transition and rare earth elements Applications for transition and rare earth elements include optical emitters and, in particular, electroluminescent components. Erbium, as noted above, is used in generating lasing effects. All these components are produced by inserting optically active ions into insulating or semi-conducting matrices. These ions are the origin of fluorescing phenomena and bring into play either transitions between flat 4f layers (rare earths) or, principally, alterations in vibrational states (transition elements). Matrices can be doped either during the growth of layers by epitaxy or by ion implantation. The rest of this Section will deal with certain properties utilised in components, which in turn are detailed in Chapter IX on optoelectronic emitters.

1 Electroluminescence in passive matrices (see also Chapter IX, Section VII) Shock ionisation, brought about by collisions of carriers on a lattice, can generate carriers at excited levels on luminescent centres. Until recently, these centres were typically based on phosphorescing inorganics (for example, ZnS activated by Mn 2+) and were introduced either using a dielectric bonding material (Destriau 's effect) or in a structure permitting reinforcement of charge injection (such as MIS structure, Schottky junction and thin multi-layer films corresponding to ACTFEL structures). An example of the direct relationship between the crystal effect and transition metal ions is that of manganese inserted into ZnF2: the former produces an orange emission centred at 580 nm and the latter has emissions shifted towards longer wavelengths (towards yellow) with the principle emission at 585 nm (2.12 eV) along with others detailed in Figure VII-9 [Bren 99, p706].

172 Optoelectronics of molecules and polymers Mn2+ excited levels

I

2.12eV

!

Fundamentallevel

Figure VII-9. Emission lines of Mn2+ insertedin ZnS.

2 Insertion into semiconductor matrix It has been shown that for erbium to undergo photoluminescence, the band gap of the host semiconductor must be larger than the energy (0.805 eV) associated with the wavelength (1.536IJ.m) of the transition 4113/2 --+ 41 15/2 . In addition, the intensity of luminescence decreases with the excitation temperature, just as much as the decrease in the semiconductor gap; at ambient temperature, the emission is only detectable for semiconductors with a gap superior or equal to that of InP at 1.25 eV. It should be noted that the concentration of doping ions can modify photoluminescence intensity, which reaches a maximum at a certain concentration. In the example of erbium this maximum is found at approximately 1018 ions erbium cm- 3 when implanted in a layer of InP, although the value is independent of the host matrix, be it InP or GaAs. At low ion concentrations when ions are independent of each other in a matrix, in general, luminescence increases steadily with an increase in ion concentration, however, at higher concentrations when ions are in closer proximity, then excitation energies which would otherwise be used for radiative emissions can be lost to transfers between ions. The result is that above a certain concentration (often of the order of 0.1 to 10 %) the luminescence decreases. The above process corresponds to a crossed relaxation. Co-doping can result in an increase in intensity of luminescence. For example, silicon doped with erbium shows a two fold increase when doped with erbium and oxygen. This process is recognised to be due to the presence of acceptor levels on oxygen 0.35 eV higher than the highest point of the silicon valence band . Recombinations in the forbidden silicon band with these localised levels is equivalent to an energy of 1.12 - 0.35 = 0.77 eV, a value of the order of that required to excite Er 3+ ions (transition 4115/2 --+ 4113/2 of 0.805 eV) . Luminescence dependent on the direction of an applied polarisation (tension) can be obtained by inserting ions in zones of pn junctions. Possible variations are : •

using direct polarisation where ions introduced into a p zone allow radiative recombinations from electron-hole pairs and the resulting radiation, as virtual phonons, excites the inserted luminescent ions. If recombinations are

VII Optical processes in molecular and macromolecular solids

•

173

non-radiative, then the de-excitation energy can nevertheless be transferred to a luminescent ion which in turn can re-ernit photons with energy less than or equal to the received energy; and using inverse polarisation, the excitation of rare earth or transition element ions results from shocks due to hot carriers (in the Zener characteristic zone) such as electrons injected from zone p into zone n, in which they undergo collisions. The n zone is thus preferentially doped.

3 Light amplification: erbium lasers In systems using fibre optic transmissions, losses during propagation necessitate the periodic use of signal amplification along the length of the fibre. Initial designs, which were relatively expensive, used transducers to transform the optical signal to an electronic signal, which was amplified, and then back again to an impulse optical signal. A more modern technique though is to use silicon optical fibres doped (over distances of around 30 m) with a low concentration of 8rH cations, as schematised in Figure VII-lO-a. Along the distance, Er H cations are excited using a semiconductor emitted laser beam of wavelength typically between 1.48 and 0.98 urn and a 'pumping' intensity that decreases on going through the doped zone, described in Figure VII-l O-b. When luminescent impulse signals traverse the doped and 'pumped' region of the cable, they receive an energy due to the 8rH ions, as illustrated in Figure VIl-lO-c. Radiation of wavelength A = 0.98 urn excites 8rH cations from their fundamental state 4115/2 to their excited higher state 4111/2, which in turn rapidly relax without radiation to the 4113/2 level. This process is one of 'pumping' and the laser used acts as a 'pump' . Radiation with A = 1.48 u.m excites 8rH ions directly to 4113/2 which has a long lifetime. An impulse at A ~ 1.48 urn traversing the system empties all these excited levels so that the pumping energy is added to the luminescent impulse signal in such a way that it conserves its coherence.

=1

silicon fibre Ca) lOne doped with erbium

,

pump laser intensity

Co o o o o o o

: o o

di.otance

Figure VII·I0. Zone (a) shows silicon fibre doped with erbium and subject to a 'pump' laser to amplify luminescent impulses (c).

174 Optoelectronics of molecules and polymers -,

rUm-radiative transition -, "--,

E

::l. 00

~

4115/2

-,

E

::l. 00

'1;

(fundamental state)

Figure VII-H. Laser pumping overthree levels.

IV Molecular edifices and their general properties [Atk 00] 1 Aide-memoire: basic properties As we have seen in Chapter 1, macrostructures consist of molecules of atoms bonded by intramolecular forces . The solidity of the edifice relies on intermolecular forces . Molecular spectra are richer than those of atoms due to the possible rotations and vibrations. However, the energy of these rotations and vibrations, which is of the order of flE f ~ 0.005 eY and flE y ~ 0.1 eY, means that associated spectra are not in the visible region, a region essentially explored for electroluminescent and photovoltaic components. However, it is worth noting that while only electronic spectra have energies t.Ee ~ 2 to 6 eY, tied to electronic transitions at the interior of molecular orbitals, these relatively large energies can also excite rotational and vibrational levels to the point where electronic spectra are close to these absorption bands and can become very complex. We shall define electronic terms for molecules after a brief introduction using simple systems of diatomic molecules.

a Heteronuclear diatomic molecules In the example of a diatomic molecule, the axis which joins both atoms is denoted Oz. This axis of symmetry is such that the Hamiltonian remains invariant during rotation about the axis. The result is that L, and H are commutative such that they can have the same system of proper functions. We normally introduce an absolute value Ait = IMIItprojecting kinetic momentum over the axis of symmetry Oz. With M = 0, ± I, ±2, .. . we have A = 0, 1, 2 . .. which , respectively, correspond to states b , Il , fl . . . . In fact, when A i= 0, for the same energy there are two possible states . They are distinguished from one another by the direction of the projection of kinetic momentum at Oz, and an inversion of sign following a symmetry operation on a plane containing the Oz axis (L, is not invariant with this symmetry operation 0) which has invariant energy during the operation. The latter invariance is due to symmetry of the molecule on this plane, and assures the conservation of the Hamiltonian. This degeneration

VII Optical processes in molecular and macromolecular solids

175

explains the choice of A = IMI to characterise the energy states of the molecule, rather than M for which the two values would give the same value of energy when M i= O. It should be noted that at the particular case of A = 0 ( ~s tate), corre sponding to when the kinetic momentum projection is zero , Lz and 0 are commutative and are can have the same system of proper functions with H. As two successive applications of the operator 0 bring s the system back to its starting point, the proper values of t are such that t 2 = I with t = ±I (see comment below ). ~ states can give rise, in fact , to two states ~- or ~+ , depending on whether or not the wavefunction changes sign on application of the operator O. Just as in an atom , for its part the composition of all electron spins in a system yield a total resulting spin of value S, and the molecular term or ' multiplicity' is thus denoted carrying the value (2S + I) at the upper left hand comer of the letter defining the value of A.

Comment: brief look at the form of proper values associated with operator I which transforms variable x to -x Looking for proper function s of I such that I'!r(x) = t'!r(x) in which t is the proper value, also with (defining property of I) I'!r(x) = '!r(-x), so that l'!r(x) = t'!r(x) and I'!r(x) = '!r( -x) leads to multiplying the left hand side of the two equal parts by I so that: I(I'!r(x» = tl'!r(x) = t 2'!r (x) = I, '!r (-x) =

'!r (x) .

Thi s gives t 2 = I, that is t = ± I , which results in '!r (- x) = ±'!r (x) . Thi s demonstrates that proper functions associated with I are either symmetric, as in '!r+, or anti symmet-

~2 ddx 2 + V(x) ric as in '!r - . In addition, if H(x) is a Hamiltonian such that H(x) = - zrn with V(x) = V(-x), then H(x) = H(-x). Thus I H(x) = H(-x) = H(x) and the Hamiltonian is invariant with respect to I; it can commute with I and have the same proper functions as I and the proper functions of this Hamiltonian are symmetric or anti symmetric (most basic courses in quantum mechanics can give comparable notes ). 2

b Homonuclear diatomic molecules Here molecules of the type A2 are considered, where A is an atom on the second line of Mendeleev's periodic table. In these discussions, the carbon-bond-carbon system is deemed the most important, due to its centrality to the nature of small molecules and polymers, whatever the type of bond. As in Chapter I (and in Appendix 1-1), we can consider two types of molecular orbital s: • •

a -orbitals which have the straight line joining the two covalently bonded atoms as an axis of symmetry (for example, axis Oz); and n-orbitals which have a plane of symmetry which includes the axis Oz joining the two atoms and is perpendicular to an axis normal to Oz such as Ox or Oy.

176 Optoelectronics of molecules and polymers If after a symmetry operation with respect to the centre of symmetry of the molecule, the orbital sign does not change the molecules are termed as having evenparity and are denoted g for gerade (from the German word for even). If the same operation results in an inversion in the same sign, then the molecule is denoted u for

ungerade. In Figures 9 and lOin Appendix A- I (in which it is the axis Ox which joins atom centres , although this changes none of the reasoning) we can see that: • •

bonding o-orbitals have even-parity and are denoted cr g , while antibonding o-orbitals have odd-parity and are denoted cr~ ; and bonding n -orbitals have odd-parity and are denoted n u, while antibonding n-orbitals have even-parity and are denoted

n;.

We can also note that symmetry operations with respect to the central symmetry change r to -r. This operation is called an operation of parity (I), which is such that the homonuclear molecule rests invariant under its effect.

2 Selection rule with respect to orbital parities for systems with centre of symmetry In classic terms, the emission of a photon following an electronic transition is considered due to a type of electric dipole oscillating the atom or molecule (see also J.P. Perez, Masson 1996, Chapter 20 or a comparable textbook on electromagnetism or dipole ray theory) . The operator associated with the dipole momentum of a structure of atoms or molecules can be given in the form: n

M(fl ... rn ) = (-e)

L:;i, jce ]

in which ri locates the position of electrons. This operator is of odd-parity as it changes sign when all ri are transformed to -rioIts matrix element (transition dipole moment) between states 1 and 2, characterised by wavefunctions "'I (rl, . .. , rn ) and "'2(r1 , .. . , n ) , is of the form:

r

We can therefore remark that: •

•

under the effect of a transformation associated with an inversion operator I which transforms ri to -ri (and also therefore 1\1: to -1\1:), and are transformed to e I'" 1 and £2 "'2 with e 1 = ± 1 and £2 = ± 1: under the effect of I, d also becomes - £1£2 d: and in systems which have centres of inversion (symmetry), d is invariant under the effect of a transformation associated with the inversion operator I.

"'I "'2

VII Optical processes in molecular and macromolecular solids

177

The upshot is that for systems with centres of inversion then d = -£\ £2 d and -£\ £2 = I if d =1= 0, and for systems such that £1 £2 = I (i.e. systems which conserve the same orbital parities during transition from state I to state 2), obligatorily d = 0, i.e. it is a forbidden transition. For complexes which have a centre of symmetry, the selection rule (called Laporte's rule) indicates that the only electronic transitions allowed are those which implicate a variation in parity, that is those of the type g ~ u and u ~ g are allowed, while transitions g ~ g and u ~ u are not. transitions are allowed . We can also note that if the We can see here that u ~ centre of symmetry is broken (for example, by vibrational effects) then this rule is no longer respected and transitions g ~ u and u ~ g become possible. These transitions are called vibronic transitions.

Jt

Jt;

3 More complicated molecules: classical examples of existing chromophores In this Section we shall present some examples of various types of chromophores, which have groups of atoms on which transitions are localised.

a Transition metal complexes Introduction: some notions on ligand field theory Transition metals have an incomplete d orbital and in order to interpret the origin of the optical properties of the complexes of these metals, which consist of the metal tied to surrounding ligands, molecular orbital theory has been adapted under the term ' ligand field theory ' . This theory takes into account the highly symmetric structures about the central metal ion that these complexes often present. The ligands, when for example carrying a pair of free electrons localised towards a central atom, act as centres of electrostatic repulsion which force back the approaching d electrons of the metal atom . Two types of orbitals associated with d electrons can appear : ct

• •

those which are directed at the ligands and are denoted e g; those which are directed into spaces in between ligands, and for example are denoted t2g .

d

'=='===tZg Figure VII-I2. Energy level separation according to ligand field theory.

178 Optoelectronics of molecules and polymers Given the presence of these electron repulsions, an electron in the first type of orbital has a disfavoured potential energy with respect to an electron in the latter orbital. The stabilities of these sorts of complexes can be explained by the approach of ligands to the central metal cation resulting in a decreasing energy of the system, due to favourable interaction energies between the cation and those pairs of free electrons mentioned above. In addition, we have also seen that the energies of t2g orbitals are favourable with respect to eg orbitals . The orbitals between the central cation and the ligands are filled, first of all, at the t2g orbital, into which we introduce unpaired electrons. The filling up of orbitals continues with respect to the value of b.. . At low values of b.. , Hund's rule results in non-paired spins (S being the maximum) and generates the filling up of the eg level, as the reduction in energy due to Smax, which minimises electronic repulsions , is energetically more advantageous than the increase in energy associated with b.. . However, at high b.. values, Hund's rule is no longer sufficiently relevant with respect to decreasing energy before the energy jump b.. and we obtain a pairing of spins in t2g levels. ~ Transitions in transition metal complexes Transition metal complexes are often highly coloured. As we have just seen, ligands separate d levels into two subgroups, for example eg and t2g in the case of octahedral complexes , which have a transition corresponding to around 500 nm that is a priori forbidden (as is d - d and b..l = 0 for isolated atoms following Laporte's rule for centre-symmetrical systems). However, it becomes possible due to vibronic transitions in the octahedral complex . Charge transfer transitions can also occur. They correspond to the transfer of an electron from the central atom to the ligands or vice versa. An example is that of the purple colour of the permanganate ion MnO; which is interpreted as being due to a transfer from the ligand to the metal.

b Molecules with double bonds a Conjugated molecules For radiative transitions due to the transition of an electron from a n-orbital to a n" -orbital (absorption) or from a n* -orbital to an-orbital (emission) in: • •

an isolated non-conjugated double bond the gap is of the order of 7 e V; and a conjugated double bond, that is one which makes up part of a conjugated system, packing energy levels to an increasing degree with conjugation, is such that the gap is bathochromically shifted (increase in wavelength) towards the visible region.

The transition is called a n - n" transition. (It is worth noting that the symmetries of n-orbital (u) and rt" -orbital (g) permit proper accordance with Laporte's rule for centre-symmetric systems) .

VII Optical processes in molecularand macromolecular solids

179

~

Free electron pairs In compounds containing carbonyl groups (>C=O) which are active at 290 nm, the transition is due to non-bonding electrons situated in a free pair on the oxygen. In a transition called a n-n" tran sition , one of the electrons is excited to a vacant n"-orbital.

V Detailed description of the absorption and emission processes in molecular solids: role of electron-lattice coupling (electrons-phonons) 1 Electron-lattice coupling effects during electron transitions The titled coupling effects, described in Chapter IV, give rise to the fine structure of electron transitions due to lattice conformations, characterised by positions of atoms determining the system configuration for each electronic level. As detailed in Section IV-I, with respect to energies of optical tran sitions (typically in the optical domain, that is from 1.5 to 3 e V, equivalent to the order of size of organic solid bands), vibrational energies associated with various lattice configurations are negligible. These latter energies give rise to absorption and emi ssion spectra in the near infra-red . Notably, electron-lattice coupling can yield molecular electronic spectra con stituted by a collection of lines called emi ssion or absorption bands . Detailed here are tran sitions in conjugated materials, as schematised in Figure VII-13 . The overlapping of pz orbitals from adjacent sites give rise to molecular rt-orbitals, and with an electron coming from each site, the lower half of these molecular orbitals are filled, forming bonding n-orbitals, and the upper half which would otherwise yield antibonding n-orbitals, remain empty. Following excitation, the electron transition rr ---+ rt " is accompanied by a change in electron distribution and therefore a change of forces and equilibrium distances between neighbouring atom s. The transitions between fundamental and excited electronic states are vertical and follow the Franck-Condon principal. Thi s is ju stified by the fact that the mass

excited cctronic sta te

) 01

Fundamental electronic state

Conllgum tlon

co-ordinate Figure VII-l3. Configuration of excited and fundamental electron states.

180 Optoelectronics of molecules and polymers of an electron is considerably less than that of a nucleus and any movement of electrons (transitions, change in energies) occurs during extremely short time intervals (of the order of 10- 15 s) with respect to the times required for a nucleus to move (approximately 10- 13 s). We can see, qualitatively, that the height and thus the energy of electron transitions depends on the values of the configuration co-ord inates and vibration levels. From a quantitative perspective, selection rules will can be used to determine absorption and emission spectra.

2 Selection rules and allowed transitions In the adiabatic Born-Oppenheimer approximation, in which movement is slow and the movement of nuclei is negligible with respect to that of electrons, and assuming rotational energies to be negligible, the energy of a molecule can be written as E, = E, + E, with wavefunction "', = "'~ "'v in which the indices e and v indicate electron and vibrational components. The electron wavefunction can be written "'~ = '"e'" s by separating the spatial co-ordinates of electrons and their spins . This is RussellSaunders coupling, corresponding to a weak spin-orbit interaction energy, and the separation of spin and orbital movement can give the wavefunction as a product of two wavefunctions, each specific to each type of movement. Using an electron dipole approximation, the probability of a transition between initial (i) and final (f) states is thus given in the following form : ~

2

2

2

Pir ex: I("' eilMI"'er ) I I("'vii "'vr}I I("'si I"'sr} I

in which Mis the dipole momentum operator. Thus, allowed dipolar transitions are such that: •

•

tr = ("'eilMI"'er) t= O. If the dipole momentum operator is of odd-parity (M = e r) , then tr will be non-zero only if the states i and f are of opposed parity. This is found for example with bonding and antibonding orbitals as detailed in Section IV-2 (HOMO and LUMO orbitals - see also Figure VII-l3); and allowed electron transitions between different vibrational levels have intensity increasing with the factor ("'vil"'vr), which can be noted more simply as ('"vii '"vr) = ("'vi'"v') = Svv' and represents the importance of the superposing vibrational wavefunctions of initial and final states. ISvv' 12 = I ("' v I"'v' ) 12 is called the Franck-Condon factor.

The vibronic levels (due to electron-phonon coupling) appear just as well in absorption as in emission spectra (Figure VII-14) . Because of the role played by the Franck-Condon factor, the most intense transitions are those which arise from the maximum zone of the vibrational wavefunction and arrive at points of the wavefunction which are the highest possible given the vertical condition applied to the transition. In absorption this means a departure from the 'bell-shape' of the Gaussian curve of the fundamental state towards the region of the curve repre senting the oscillating excited state. In Figure VII-14 (a for absorption, b for emission), increasing probabilities of vertical transitions are represented by increasingly thick lines. These probabilities are indicated in the amplitudes of absorption and emission spectra (Figure VII-14-c).

VII Optical processes in molecular and macromolecular solids

181

s, - , ,.- ... , - _'" v'= .. ,,';; 2 ", ," ' 'y' : I

" ',

' =0

So

(a ) i

. / .I

• \

i

... . bsc rp tle n ,;'" ~ ')

'

i

i

,

(e)

i

i

\

/

/; /

i

~ni"'lon(S,-.s.)

/

/

•

i

i

.

!

i

i

I

Figure VII-14. Vertical transition intensities in: (a) absorption; and (b) emission; (c) resulting absorption and emission spectra.

In addition to the transition rules concerning the orbital movement of electrons, the selection rule for spin needs also to be taken into account in that (1/Jsil1/Jsf) =1= 0. This condition comes from the form of the electron wavefunction, which was initially written 1/J~ = 1/J e 1/Js on assuming that spin-orbit interactions are of a low energy (Russell-Saunder's coupling). Thus, we should have (1/Jsil1/Jsf) =1= 0 which means that spin states should be identical and S should be conserved during transitions. The initial and final states must be of the same multiplicity i.e. being equal to 2S + 1 with the quantum number M, such that M, = -S , -S + 1 .. . 0 . .. S - 1, S). Thus in the absence of spin-orbit coupling, tran sitions between singlet (S = 0, denoted S) and triplet (S = 1, denoted T) states are forbidden . It is for this reason that in Figure VII -14 transitions are between singlet states.

3 Modified Jablonsky diagram and modification of selection rules: fluorescence and phosphorescence a Yields from radiating states

=

°

=

For a singlet state S and M, 0, there is only one state represented by the spin wavefunction 1/JS=O.MS=O = 1/J0 o. However, for the triplet state where S = I

182

Optoelectronics of molecules and polymers

and Ms = -1 , 0 , 1, there are three pos sible states corresponding to three spin wavefunctions 0/1-1, 0/10 and 0/ II . The statistics of spin therefore indicate that there are three triplet states for each singlet state. For electroluminescence, following the random injection of charge carriers, we should obtain the same distribution of states at excited levels and the return to the fundamental state (which is a simple singlet state as electrons at this level cannot be distinguished by their spin state , which must be different) does not occur radiatively except for electrons leaving excited singlet states. The three triplet excited states do not, a priori, recombine radiatively with the only singlet fundamental state, leaving at best only 25 % efficiency with respect to radiative efficiency.

b Possible deviation from selection rule In reality, it has been empirically observed [Pop 82] that transitions between states with different multiplicities are 103 to 105 times slower than those between states of the same multiplicity. The correspondingly weak tran sitions are due to residual, weak spin-orbit coupling which exi st in all molecules (denoted by the coupling term jj and detailed further in Appendix A-8, Section II). With increasing atom ic weights though these transitions can become more intense with mixing of orbital movements and spin. In organic molecules, replacing hydrogen atom s with tho se of bromine or iodine can reinforce forbidden 'inter-system' transitions. The different tran sition s can therefore be represented with the help of the modified Jablonsky diagram shown in Figure VII-15. Finally, the excitation of an electron from a rt-orbital to a n "-orbital result s inbeyond changing the equilibrium geometry of the mole cule-a first excited state ExchanJ:c of th rmal coer!:)" (p h OOOM) ; non -ra d iat ive t nsitlon:

l

inter- S)' ·tem crossing'

Absor pt

I

Phospbu -

phosphore. ccnce

-+-+~----Lt+-l-r,escl'nce

(Rudia tiw -+....L..-------X+-l tra nsition not "'-~-----........, completely forbidden: spin-orbit coupling breaks lectlon nile)

: . Singlet

( b)

(a)

Figure VII-IS. (a) Jablon sky diagram indicating characteristic fluorescence and phosphore scence transitions; and (b) inter-system cro ssing from singlet to triplet states.

VII Optical processes in molecular and macromolecular solids

183

which can be (Figure VII-15-a) either a singlet state Sl (which with vibronic levels yields levels SIO, Sll, S12, Sl3 .. .) or a triplet state TI (which with vibronic levels yields levels TlO, Til, T12, Tl3 . . .) and that in emission there are two possible mechanisms: •

•

fluorescence is practically instantaneous after the molecule has been excited by light. The process starts with the excited molecule descending through its vibrational levels in an internal conversion process detailed in Figure VII-15-a in which energy is given to neighbours by collisions. On reaching the lowest vibrational state, of the same multiplicity (S 10) as the fundamental level (Soo), the molecule spontaneously emits a fluorescing ray during subsequent transitions , which respect classic selection rules (see Figure VII-14-b). The intensity of the emitted rays depend s on the Franck-Condon factor and the energy of those same rays are less than that of absorbed rays, as detailed in Figure VII-14-c in which we can see that emitted rays are red-shifted, have the longer wavelength s and have undergone a ' Stokes shift' or Franck-Condon displacement; or phosphorescence which occurs a considerable time after the absorption event. This emission results from the presence of excited triplet states . At the intersection of the two configuration curves , shown in Figure VII-15-b, the two states have the same geometry and for molecules which have strong spin-orbit interactions (for example those which contain atoms , typically, heavier than mercury) the molecule can undergo a reverse in orientation with respect to the spins and pass into a triplet state . After a descent through the vibrational levels to the lowest vibrational fundamental level of the triplet state, the energy remains trapped due to Russell-Saunders coupling which forbids triplet-singlet transitions. As we have already seen, intervention by non-negligible spin-orbit coupling can modify the selection rules (as we can no longer separate spatial co-ordinates of electrons and their spin and thus write the electron wavefunction simply as \)I ~ = \)Ie\)Is) so that the molecules can slowly emit.

Electroluminescence brings into play the same transitions as those detailed above . We shall now go on to see how quantum yields can be improved in these systems.

4 Experimental results: discussion a Empirical results Figures VII-16 and VII-17 show spectra obtained for absorption and electroluminescence of Alq3 and poly(para-phenylene vinylene) (PPV), respectively. In both cases we can see a displacement of the emis sion to the red. Alq3 shows both spectra to be of approximately the same size. PPV, however, shows an absorption spectra larger than that of emission, and Figure VII-I 7 is less symmetrical than portrayed by the theory shown in Figure VII-14-c. This could be due to the distribution of the conjugated lengths of the polymer, with shorter conjugated segments contributing to a shift towards higher energies . All conjugated segments, whatever their length,

184

Optoelectronics of moleculesand polymers

- Encrg)'

Figure VII-t6. Absorption and electroluminescence spectra of Alq3.

participate in absorption, resulting in a wider spectrum, and the preferential diffusion of excitations to lower energy segments gives rise to these emissions [Gre 95].

b Discussion a Configuration diagram extended to polymers Considerable discussion has been made concerning the application of a model representing electronic spectra, shown in Figure VII- 14, to conjugated polymers which a priori present very delocalised orbital bands. This is an issue of extending a description of states with the help of localised molecular orbitals (small molecules which display essentially discreet energy levels) to polymers. It is possible because of the effect of electron-lattice interactions (see Figures IV-6 and IV-7 for the same systems in Figures VII-13 and Vll-14) found in conjugated polymers which give rise to sufficient locali sations of states during radiative emissions (excitonic states) . See also discussion in Section VI-2 concerning the nature of excitons in polymers where empirical results indicate, overall, that the

1.8

2.3

2.8

3.3

3.8

Encrgy (cV)

Figure VII-t7. Absorption, photoluminescence (PL) and electroluminescence (EL) spectra of PPY.

VII Optical processes in molecular and macromolecular solids

185

argument is correct. However, in certain cases , difficulties with this approach should not be ignored. ~ Extendin g excitations: notion ofexcitons In the discussion j ust above, an excited electron was seen as being localised on a single molecule, however, this is not strictly true when considering recombinations, a reasoning also generally applicable to polymer s. In reality, in polymer s as in small molecules, the generation of an electron-hole pair, by photoexcitation or double electronic injection, results in an excited but neutral state with a limited , finite life time. This state is termed an exciton and is constituted of an electron and a hole paired by excited energy states within the limits of permitted bands (LVMO and HOMO bands, respectively). The occupation of these excited state, the LVMO by the electron, the HOMO by the hole, is termed a nonrecombined exciton and has unpaired electron spins, while recombination results in a double population at the HOMO level (obligatorily a singlet state tied to paired carriers). In an organic crystal, recombination of electron and hole is generally Coulombic in origin , while capture is termed Langevine. In effect , the Coulombic bonding energy between an electron and hole is greater than therm al energy ([q2/ 4m:r] > kT) when r < rc = (q2/4m:kT). As for organic solids, E is small ( ~3) and rc is large ( ~20 nm), so with surface area of efficient capture defined by Or = JU~ (of the order of la-II cm 2), capture is relatively easy. A simple calculation of charge distributi on within an organic solid shows that all charges are found within the same Coulombic sphere as opposed charges. A bonded electron -hole state corresponding to a neutral quasi-particle (exciton) forms in such a way. In addition, given the poor mobility of carriers in organic solids, the carriers can only comb ine through processes deta iled in the above Sections I to 3. However, as we have seen, depending on the solid, the excited electron-hole pair can be localised on the same molecule or over different molecules. In addition, important effects can result when the excited molecules arc close together (as in aggregates).

VI Excitons 1 Introduction In general term s an exciton can be defined as an elementary excitation of a poIyelectron insulator [Vet 86]. Thu s the excitat ion of an electron from a valence band generates an excited electron-hole pair, which as a quasi-particle is subject to a number of interactions. These include Coulombic interactions (described in previous Section) between the electron-hole pair and other interaction s caused by interatomic , intermolecular and excited system dipole-dip ole forces. These interactions effect energy levels and the optical transition s in the solid. Simpl y put, we can suppose that an electron and a hole generated by excitation mutually attract to form an assoc iated, bonded state called an exciton much in the same way as an electron and a positively charged default are tied within a solid by an electrostatic force.

186 Optoelectronics of molecules and polymers

e e, ... _e_ e' e e ,/e e e '.

•e. e. ...

-""' )I

,_/

\

\

e \e... e e , . .e e . . . '• • •

(a)

(b)

(c)

Figure VII-IS. Various excitons represented: (a) Frenkel exciton; (b) Wannier exciton; (c) chargetransferexciton.

Depending on the nature of the solid, the excited state of electron and hole pair can be localised on one or more molecules. The former is called a Frenkel exciton, and is detailed in Figure VII-18-a. For an electron and hole separated over several molecules, the result is called a Wannier exciton (Figure VII-18-b). The intermediate between these two is the charge transfer exciton, where electron and hole are on adjacent molecules, as shown in Figure VII-18-c .

2 Wannier and charge transfer excitons In solids with large permitted bands (size B) such as semiconductors, electrons or holes exhibit high mobilities (u) with Il- proportional to B, as detailed in band theory for covalent solids. Holes and electrons can therefore easily separate giving rise to low interaction energies. Given that an electron and hole pair, with energy levels schematised in Figure VII-19-a, are: •

•

buried within a continuous medium with permittivity given as f = fOf r where f r is relatively high for a reasonably well conducting medium (quite high mobility) ; and trace an orbit around one another in the material; thus their bonding energy (En) can be evaluated by using the analogy of semiconductors where we introduce charged states due to acceptor or donor type impurities. The energy En is thus given by the expression:

• •

derived from a hydrogen based model for which quantified energy levels are of the form -m*e4/32n 2 f 2fi. 2 n2 where m* is the effective mass of the system; and is such that for a limited state, non-bonded (n -+ 00) with the electron-hole system, the energy is equal to Eo (material gap) which effectively corresponds to the liberation of these two particles from conduction and valence bands .

VII Optical processes in molecul ar and macromolecular solids

51

:---

187

-+-

~O+ l.------ ~± (a) Fundamental state

Wannier exciton

(b) Fundamental state

Frenkel exciton

Figure VII -19. Representations of excited states in: (a) classic semiconductor band scheme (Wannier exciton); and (b) molecular state with discreet levels (Frenkel exciton).

On using the highe st point of the energy band to define the position s of bands due to bonded excited exciton state s (En), which are below the minimum of the conduction band , (situated at EG with respect to the highest point of the valence band) where the electron and hole attract one another with the Coul omb ic potent ial 2 VCr) = -e /4mr , En = EG

*

4

memh m e 2 1 1 2 l ' with m" = - -32Jt E6Erh nme + mj,

The radiu s of the exc iton (rn ) can be evaluated using the electron-hole distance again for a sys tem based on hydrogen and is therefore of the form rn = n2(4Jt EoErfl.2/m*e2). The distance rl at n = I, the lowest energy level for a Wannie r exci ton and thus the most bonded exc iton state (like a Is orbital) , is the shortest. From these expressions for En and rn, we ca n see that bond ing stre ngth is dependent, esse ntially, on two parameters: • •

the dielectr ic permittivity of the medium: the larger it is, the wea ker the electronhole attraction; and the effective mass of electron-h ole pairs: the smaller it is, the grea ter the electronhole distance (and by consequence the harder it is to retain the exc iton).

Thu s for semiconductors which exhibit a large permitted band , high permittivitie s and charge mobilities f-.l (with a low effective mass m" as, classically, f-.l = qt yrn"), only low bond energies ca n appear (Wannier excitons). For semiconductors with an indirect gap, excitations of the lowest energy are forbidden as they require phonon intervention. For example, Wannier excitons have only been obse rved in semiconductors such as Ga As with a direct gap. For polymers, Wannier excitons have been theoreticall y envisag ed in PPY [Gre 95]. The bonding energy of the excito n was found to be around O.4eY within an ellip soid geo metry with main axis 2 nm and secondary axis 0.4 nm. These values are represent ative of molecul ar excitons with a high degree oflocalisation on a chain under strong electron-lattice interac tions (and electro nic and vibrational state coupling). Polymers with a wide gap (insulato rs) must, neverth eless, present a weaker intrachain delocalisation and inter-c hain interactio ns ca n be com para tively grea ter. Charge

188

Optoelectronics of molecules and polymers

transfer excitons (intermediate to Wannier and Frenkel excitons) are possible. The optimisation of interchain contacts can result in excitons termed excimers (excitons shared over several identical molecular units) or exciplexes (excitons shared over 2 or more different molecular units) . The bonding energy of a charge transfer exciton is given by: ECT = Ip - EA P - C in which: Ip is the donor ionisation potential (hole site); EA is the acceptor electron affinity (electron site); P is the polarisation energy of the lattice by the electron-hole pair; and C is the Coulombic attraction energy of the electron-hole pair. Using this method, the correct bonding energies for tetracene were calculated with a Coulombic energy for a hole placed at the origin with an electron centred on neighbouring molecules [Wri 95, pI09].

3 Frenkel excitons In an inverse scenario to that detailed above, in molecular crystals, the formation of Wannier excitons necessitates a radius of excitation (rn ) considerably greater than the intermolecular space available . For materials which are good insulators, tr is low-along with mobility-due to poor contacts between molecules, and electron hole interaction energy is great and rn is low. The result is that the exciton remains localised on a single molecule, corresponding to a Frenkel exciton [Kao 81]. In molecular crystals, the covalent bonds between atoms that make up the molecule are much stronger than intermolecular Van der Waals bonds. As we have already seen in Chapter III, Section III-2, transitions between electronic levels of a practically isolated molecule in a diluted state and one in a condensed, solid state are only weakly displaced in frequency. On the basis of an excited state on a strongly bonded host molecule , Frenkel excitons were generally used to explain luminescence phenomena in molecular crystals . The corresponding energy scheme is detailed in Figure VII19-b. Frenkel excitons thus correspond to a strong bond approximation with the excitation localised on the same molecule (see Section V) or on an adjacent neighbour. Frenkel excitons have been observed during n-n" transitions in aromatic molecules such as anthracene. Another example is that of excitations in ligands fields of d electrons, observed for example in nickel oxide [Cox 87, p225]. While the particles that make up the electron-hole pair are bonded to one another on the same lattice site, together they constitute a quasi-particle which can move through the crystal by transferring energy to neighbouring sites. This representation of energy migration is confirmed by the fact that in crystals which contain impurities, the latter can trap excitations. It is in this manner that anthracene (which fluoresces on optical excitation), once doped with several parts per million of tetracene, sees its own fluorescence decrease and that of tetracene appear. This effect demonstrates that tetracene efficiently traps excitons, which can move across relatively large distances without relaxing. Given the relatively low concentrations of impurities, the distance covered can be of the order of a hundred molecules or more.

VII Optical processes in molecul ar and macromolecular solids

189

The movement of excitons can be due to overlapping of orbitals bet ween adjacent sites. However, it should be noted that durin g the permitted rr-rr" transitions, it is the electrostatic interaction tied to the dipol ar momentum which gives rise to the greatest coupling energ y between fund amental and excited states . This term is expressed in the form 1L ~ /R3 in which lLij is the dipolar momentum for the transition between fundamental (i) and excited ( j) states . For adjacent molecules of different types, the exci tation energy transfer mechanisms with long (dipolar interactions) and short (orbital overlapping) action radii will be detailed, along with Forster and Dexter transfers, in the following Secti on 7. The se transfers help explain the interest in the use of optical doping of fluorescent molecules especially for organic LEOs and can also help get round selection rule s to increa se LED yields (Chapter X). Befor ehand though we shall look at the inter action effects due to neighbourin g excited molecul es which can result in particular properties, especially for exa mple in aggregates . The se effects includ e: •

•

•

interactions between two identical molecules without bond formation (termed a physical dimer) in which discreet levels appe ar and the result, in terms of transition s, depends on the orientation of each of the molecular dipole s; interactions between identica l molecules in the same chain for which there is a breakdown of the precedin g levels, associated with excited electronic states representing Frenkel exciton wavefunctions. The resulting band is ca lled an exciton band and while it does not appear optically (due to the rule Ak = 0) there is, however, an observable change called Davido v's displacement ; and interactions betw een molecules, which in crystallographic term s are unequal (or have two different axes) and the preceding band breaks down into as many band s as there non-e quivalent crys tallographic axes in a process termed Davidov's breakd own.

In the following Section each effec t will be described, although the first will be detailed most thorou ghly.

4 States, energy levels and transitions in physical dimers a States and energy levels A physical dimer is made of two identi cal molecules which are close to one another but do not have a chemical bond . With respect to individual single molecules, there are spectroscopic modifications due to intermolecular interaction s. In order to under stand them more fully, we can write their Hamilt onian in the form H = HI + H2 + V 12, in which H I and H2 are Hamiltonians of the isolated molecules and V12 the intermolecular potential energy. Having ilr 1and "'2 as wavefunctions of the fundamental states of molecule s I and 2 (neglecting spin and vibration states), of the dimer can be approximated by the wavefunction "' f = '" I "'2 denoted eqn (4- 1). and is no more than an approxi mate solution to the problem as in the physical dimer there is also the intervention of the interaction term V 12. Give n this approx imation of the wavefunction, the energy of the fundamental state is in the form Ef = E I + E2 + ('" 1"'2 IV 12\ '" I "'2)

190

Optoelectronics of molecules and polymers

where E\ and E2 represent state bonding energies of single molecules and the latter term , W = (\jJp/J2IV121\jJ\\jJ2), corresponds to the Coulombic bonding energy of the pair of molecules in their fund amental state. In reality, the calculation in this approximation is in disagreement with observed results and it nece ssitates a slight readjustment. To do this , it is nece ssary to con sider the interaction effects of the two mole cule s I and 2 excited to equivalent \jJ f and \jJi state s, respect ively, if the molecules are the same. In the limiting case, if only one of the molecules is excited which rests on the same molecule, then V \2 = 0, and the state of the system is denoted by wavefunctions \jJf\jJ2 or \jJi\jJ\ to which there are the same, corresponding energy states Ef + E2 or E I + Ei . However, if V 12 f= 0, as is the case in physical dimers, then the electronic energ y is no longer localised at one molecule but oscillates between the two molecules of the system. The dimer is therefore not degenerate. The variation s in energy depend on the interaction energy and, therefore, the orientation of the two molecules. If the two molecules are identical then , a priori , each of these molecules has the same probability of being excited. The physical dimer once excited can therefore be developed following a linear combination using the possible wavefunctions \jJf\jJ2 and \jJi\jJ, of the initial system (single excitation on one molecule with V 12 = 0). We can therefore write : (I )

Denoting the energies of excited 'monomers' as Ef and Ei , the corresponding energies (to \jJE± state s) of the excited dimer system are in the form E± = Ef + E2 + W ' ±~ with :

•

•

W ' = (\jJf\jJ2 IV121 \jJf\jJ2) = (\jJ, \jJi IV 121\jJ 1\jJi ) being the Coulombic energy for interactions due to the charge distribution on excited mole cule I and unexcited molecule 2 (or vice versa) in a term which depends on polari sation or the polari sability of the media in excit ed and fundamental states ; and ~ = (\jJf\jJ2 IV121 \jJ\\jJi) being the reson ance interaction energ y depending on the overlap of the two molecular orbitals.

Figure VII-20 show s the energy levels for a physical dimer. We can see that W and W ' are negative and that with ~ < 0, the level E+ is lower that level E_ . The two possible transitions + and - are also shown .

b Possible transitions In fact, depending on the orientation of the two molecules [For 65] , one of the possible tran sition s + or - is forbidden , and in effect, for a dimeric system, the transition momenta between the fundamental \jJ\\jJ2 state and the two, possible excited \jJE± states are in the form (see also Section IV-2)

VII Optical processes in molecular and macromolecular solids

"'E. E.

EI*+Ez Ez*+E I (Two isolated molecules of which one carries excitation and V I Z = 0).

2~

"'E+

191

(Physical dimer excited once, V 12 ;t: 0)

E+

+ (2 isolated and unexcited molecules V12 = 0)

"'1"'z

W Dimer (V tZ ;t: 0)

Figure VII·20. Physical dimer energy levels E+ and E_ .

-

-

using M , and M2 as the tran sition momenta relative to eac h mon omer (with for example, MI = (\111 l-e(Li l fil )1wr)) : M± = ~ (fvt l ± ~h ) . For a 'top to bottom ' orientatio n of the two monomers, show n in Figure VII- 21 -a, the tran sitional moment a of the dimer can be summed (mo mentum M + corres pondi ng to energy E+ ) to give an allowed tra nsition (indic ated by full arrow), or ca n zero eac h other out (mome ntum M_ co rres po nding to energy E_ ) resulting in a forbidden tran sition (indicated by dashed arrow). A straightforward and see ming ly reas onable result is obt ained with a par allel mon om er orie ntation, as shown in Figure VlI -21-b. However, when the tran siti onal mom ent a of the monom ers are neither parallel nor anti-parallel, as de scribed in Figure VII -21-c, then the result ing transit ional momenta

M

Excited levels

,---.- f , '_ ....--1- t ,

I

~

~

M+

Fundamental level Monomer Dimer (a) 'Top to boltom' organisation: red-shift.

~~

/=rl 1

. Excited levels

, ,

'~ - ~ -

i

Fundamental level Monomer Dimer

tt

- t+

'""':,::::" (lowest )

(b) Parallel organisat ion: blue-shift.

Excited levels

,- ---.-- )' t' I

,

,

M resuhing

.l' ......

Fundamental level Monomer Dimer (c) Oblique organisation: breakdown in bands (limited here to 2 transitions).

Figure VII-21. Classification of possible transition for three classic arrangements for 2 molecules in a physical dimer (note, for simplification, W and W' displacements not shown).

192

Optoelectronics of molecules and polymers

of the dimer take on two , non-zero values to give two possible transitions of energy separation ~E = 2~, permitting evaluation of B.

5 System containing an infinite number of interacting molecules and exciton band: Davidov displacement and breakdown a Qualitative description of interaction effects due to n interacting molecules and the exciton band Instead of a physical dimer ofjust two molecules, detailed here is a chain of n identical molecules. This jump is analogous to going from studying the electronic structure of a covalent bond between two atoms to looking at a one dimensional covalent semiconductor, as in Chapter I. Instead of there being discreet levels for two bodies, there is an energy band in a 1-0 crystal of infinite length and band size dependent on the degree of orbital overlap. The discreet levels E+ and E_ shown in Figure VII-20 each give rise to a band of narrow levels which are termed crystal exciton band s.

b Linear chain of N identical molecules spaced d apart If there is only one molecule (i) excited to wavefunction '!Ii while all other molecules denoted by n =1= i remain in their unexcited state '!In, the wavefunction for the system can be written: \II; =

wi

nWn. n

In reality, the wavefunction to which a particular molecule is excited is not actually a good physical solution (as we have seen for the dimer). And as in band theory where we use a linear combination of atomic orbitals, here we must also use a linear combination of all excited states of the form \IIi which can appear. The eventual wavefunction should therefore be looked for in the form \II = c;\IIi and the resulting solutions to the wavefunctions are subject to the same considerations as those used for band theory (limiting conditions especially) which are of the form

L?=l

where d represents the distance between molecules. The function \Ilk is such that an excitation is not confined to one molecule but rather delocalised over a whole chain, and is the wavefunction of an excited, neutral but mobile state of the cry stal: it denotes the exciton with wavevector k. Just as in the LCAO method, we can calculate the energy E(k) associated with the \Ilk state. Denoting the interaction energy between two adjacent molecules by ~ and the tran sition energy relative to a system of isolated molecules (gaseous phase)

VII Optical processes in molecular and macromolecular solids

193

by Eo, we obtain E(k)

= Eo + (W -

W')

+ 2~cos(kd) .

Interaction therefore results in the appearance of a band of size 4~ (Figure VII-22-a). In reality, and from a spectroscopic viewpoint, this band should not appear as the selection rule t.k = 0 states that only a single electron associated with k = 0 can result. So, the excitation energy E is simply displaced (by (W - W') + 2~) with respect to that of isolated molecules . This is called Davidov 's displacement and can be clearly observed, for example in charge transfer components formed from [Pt(CN)4]2- . c Linear chain consisting of two types of different molecules This more complex case can be treated using the analogy of an I-D, AB type crystal with a strong bond approximation (see Chapter I, Section V). Each 'ltk function can be seen as a linear combination of all possible linear combinations such that

I

'It k

N

=" eikpd(aP 'ltPA + bPWPB ) JNL., p=]

where 'lt~ and 'lt~ represent wavefunctions for the A and B molecules excited at site p. For each value of k, and in particular k = 0, which corresponds to the optical region, there are the two possible transitions a and b (Figure VII-22-b) associated with different values of ap and bp. The interaction between two different molecules A and B results in a Davidov breakdown of the transition. This effect has been observed for anthracene, which gives rise to two forms through a translation of its unit cell. The intensity of the breakdown (WD) depends on the interactions between the non-equivalent molecules (through a plane for anthracene) and the 4~ size of each band depends on both sorts of intermolecular interactions. From a practical point of view (for example phtalocyanines [Wri 95]), the spectral peaks from a solid are displaced and spread out with respect to those from a solution. (a)

(b)

Level degenerated 2N times

Level degenerated Ntimes

Eo

E

a

-,

t

-, -,

,--'----'--

Crystal ofN molecules

Crystal of2N molecules

Figure VII·22. Energy levels of excited states for a chain of: (a) N identical molecules ; or (b) 2N AB type molecules .

194 Optoelectronics of molecules and polymers

6 Aggregates [Fav 01] a Aggregate forming systems In general, colouring agents in solution display wide and unstructured absorption spectra. However, certain ionic colo uring agents, which carry charges paired with counter ions, in sufficiently concentrated aqueous solutions show a particularly discreet and intense absorption band red-shifted with respect to the same agents alone. This band is indicative of the formation of molecular aggregates (uni-dimensional) which are generally called 'J aggregates', and have as example pseudoisocyanine. More precisely, for an isolated molecule, or one which is in a very dilute solution, the same absorption band has a frequency VI ~ 19000 em:", while at higher concentrations, this band decreases at the expense of a second band which has higher energy and is situated at V2 ~ 21000 em-I. At even higher concentrations of the same molecule, a narrow band (width D. v ~ 180 cm- 1) with a lower energy appears which is the J band and has VJ ~ 17500 cm'" :

b Origin of observed effects The above observations detail a process in which the system is not at too great a co ncentration, and any formation of dimers occurs at the expense, twice over, in the co ncentration of monomers. If the transition al momenta of the two molecules in a dimer are parallel to each other and perpendicular to the axis of the dimer-as shown in Figure VII-21-b-it is the upper level which corresponds to the permitted transition and thus also absorption. Accordingly, there is a blue-shift. Transitions to the lower level are forbidde n, so that following relaxation from the higher to the lower level, any fluorescence of the system will be very weak and will penalise any electroluminescence (Figure VII-23-a); the pair of molecules take on the co-o rdination of an ' H aggregate'( the H resembles the geo metric form). However, once the system is relaxed, the return from excited to fundamental states is improbable and the excitation (exciton) can diffuse over relatively long distance s (~ 100 nm) displaying a propert y which can benefit photovoltaic

,H aggre ates

_rL

··=:=;rr:"'IiTr+--""'l.o-n-~d-::l=lTu"",I'""on

t . L

= ='==1== W:W. .....

c

-,

0

1: ! z : "

I

~ i

~ i ~ ! i

-. I

.

two monomers

(. 100 nm )

or C"1rltatJon -

....... is. ·s I .... """ 5: ~ 51 §

=5::; -; :.e: 5~ .? t: 0 - , - t:

N I ~

.. t

-. (a)

•

Fort,idden I • • transmon

:

:Vcl) '" aL

I tl Ut l ll.' ~ nc("

...

one dirner (e[ Figure VIl-2 1-b)

, ~.~.' = ='==1== ,.... : I ~,.,

1: ' c

I

~ ', l: g '

;i; ,' I

two mo nomers

.

l:SSh-;;r1dTlT~.T.;;;--(_ lO nm)

or u
.....- 1' --r==J..........-a-I

.g t~

cCJ

cd].§~

:e~"' ::a....og -e E-51 ,g]t ... Eo = :Il.Q

0

=

'

-

S

t. ---1-:-:---one dimer (e[ Figure

-.

VIl- 21-a)

(h)

Figure VII-23. Transitions and excitation diffusion for: (a) H aggregate; and (b) J aggregate.

VII Optical processes in molecular and macromolecular solids

195

effects, where the necessary separation of hole and electron is favoured by the presence of a capable electric field (see volume heterojunctions detailed in Chapter XI). In addition, the J band which appears at the highest concentrations shifts towards low energies (red-shift as detailed in Figure VII-21-a). Aas the corresponding fluores cence is rather intense we must suppose that molecules at the centre of J aggregates (the letter J symbolises the bottom to head alignment of the molecules) are orientated so that which ever level absorbs or emits, the bottom to head alignment is retained. This positioning of the molecules is beneficial to electroluminescence (Figure VII23-b) but penalises any photo voltaic properties. Excitons can recombine easily and therefore diffuse only over short distances, leaving little chance that they will meet a centre, such as a volume heterojunction which could give rise to a potential difference permitting their separation. To conclude, in order to obtain fluorescence the parallel arrangement of molecular transition momenta are proscribed for condensed materials. However, if aggregates do form, then at least type H aggregates should be avoided, while type J (which have linearly trained dipole momenta) are preferred. In photovoltaic systems, the inverse is true: J aggregates should be avoided as they display low exciton diffusion lengths, while H aggregates are favourable to the separation of holes and electrons. We can note that the coupling effect detailed here has been observed in the n-conjugated polymer poly(para-phenylene) (PPP) in its ladder form (LPPP). In addition, a line shifted towards the yellow appears for PPP films, while a blue line is associated with isolated polymer chains. In addition, the fluorescence quantum yield for PPP goes from 0.4 in solution to 0.1 in the solid state [Lem 95] , an effect attributed to non-radiative traps in the solid, condensed state films. In the solid state, needle shaped molecules tend to orientate themselves parallel with respect to their principal axis, so that the dipolar transition moment for the lowest excited state is also directed in the same direction . Oligophenylenes, which have this structure, exhibit quantum yields in solution of the order of 10 % but extremely low yields in the solid state (less than 10- 3 ) [Sch 00] . Low quantum yields in electroluminescent diodes are generally blamed on local order. In order to increase yields, films with amorphous structures which avoid crystallisation are sought.

7 Forster and Dexter mechanisms for transfer of electron excitation energy a Possible mechanisms for excitation transfer between donor and acceptor molecules The principal mechanism for energy transfer is from an excited donor molecule (D*) to a molecule which acts as an acceptor (A) . The transfer can be written in the form D* + A -+ D + A*. It is in this way that optically doped organic films transfer an excitation (exciton) from a host molecule to an 'invited' (doping agent) molecule, in an effect which can reinforce luminescence of invited molecules to the detriment of host molecules. A system can therefore evolve by going from the initial state \{Ii = \{IO*+A to the final state \{If = \{Io+A* . The probability of such a transition can be written,

196

Optoelectronics of molecules and polymers

using Fermi's golden rule, in the form Pif ex: J(\IId H(l) l\IIf) 12 N(E) where N(E) is the density of states in the donor-acceptor system at the energy state under study. In organic materials, the probabilityof transfer of kET excitons per unit time is high when going in the direction donor to acceptor. Once the acceptor is in an excited state, it relaxes to the lowest vibrational state with a probability much greater than that associated with a return of excitationfrom A* towards a donor moleculeD. The term H(l) embraces the interaction terms between donor and acceptor which assure the passage of the excitation. These terms can be divided into two groups: • •

electrostatic interactions (Coulombic interactions due to preponderant dipoledipole interactions) which gives rise to Forster type transfers; and direct exchangeinteractionsbetween molecules adjacent to donors and acceptors (forexample,hoppingor excitationdiffusion betweenoverlapping electronclouds of adjacent D* and A molecules) which give rise to Dexter transfers.

b Properties of long distance Forster transfers (3 to 10 nm) If 1: represents the lifetime of an excitation D in the absence of a transfer towards A (the lifetime of an exciton localised on D has probability of de-excitation per unit of kD = 1/1:), the probabilityof excitation transfer per unit time via the Forster mechanismis given by:

I(R: )6

kET(R) = ~

where R is the donor-acceptor distance and Ro is the Forsterradius which is such that R = Ro when kET = ko. Dipole-dipole interactionsare non-negligible up to distances as long as 10nm in organic media, due to strong electron-lattice interactions which result in an increase in emission or absorption spectra. Spins of both A and D molecules are conservedduring Forster transitions under a conditionimposedby alloweddipolartransitions in thedonorand acceptormolecules,

-----t-

+ + (c)

Figure VII-24. Excitation transfer, between singlet states, from D* to A following the: (a)Forster mechanism; and (b)Dexter mechanism. (c)Shows state following excitation transfer according to I D* + I A ~ 1D + I A*.

VII Optical processes in molecular and macromolecular solids

197

--+-----=--a---*_ 3D*

t(s:-l)_n_~O)

Figure VII-25. Dexter transfer with conservation of exciton state (example here is triplet): 3D* + IA --+ ID + 3A*.

so we find that the following transitions are allowed: 1D* + I A --+ 1D + 1A*(ef scheme in Figure VII-24-a)

ID* +3A(T n) --+ ID+ 3A*(T m ) in which Tn and T m are, respectively, spectral terms for the molecule A in vibrational state n and the excited molecule A* in the vibrational state m. In the latter transfer, the spin configuration of the exciton is changed. In the Forster mechanism, transferred excitons can undergo a change in spin, as in the example here where the spin goes from singlet to triplet. + 1A --+ I D + I A*) is a prior forIn contrast, the triplet-singlet transition bidden. Nevertheless, it can occur because the lifetime of the 3D* can be long, and although the probability of exciton transfer per unit time (kET) from 3D* to 1A remains small , it can be higher than the probability per unit time for the transition 3D* --+ 1D.

eD*

c Properties of short distance Dexter transfers (0.6 to 2 nm) In contrast to the Forster mechanism, the Dexter mechanism does not rely upon allowed transition probabilities in donor and acceptor molecules. The Dexter transfer displays a probability in physical coherence with the proportional overlap surface of donor and acceptor molecular orbitals. It is a transfer over short distances and subsides exponentially with distance. Only the total spin of the system is conserved (D*A then DA*) during the transfer of excitation via overlapping electron clouds, and an exciton transferred this way retains its spin configuration: a singlet exciton retains its singlet state in the same way as a triplet exciton remains a triplet. In addition, the Dexter transfer is the only one which allows energy transfer from the donor triplet state to the acceptor (in the Foster transfer, the transfer of energy from donor triplet state to acceptor triplet state is forbidden due to the condition requiring spin conservation of each type of molecule involved in dipole-dipole interactions). The Dexter transfer of energy between donor and acceptor triplet states can be represented by: 3D* + 1A --+ I D + 3A* (Figure VII-25). The Dexter transfer is further illustrated in Figure VII-24-b in which there is a transfer of excitation energy between singlet states , as was the case for the Forster transfer.

VIII Fabrication and characterisation of molecular and macromolecular optoelectronic components

This Chapter describes general method s used in the fabrication of organic optoelec tronic components with two specific examples which have undergone considerable development during the last decade. One is that of the electroluminescent diode (from 'small' molecule s to give OLEDs, or from polymers to give PLEDs) and the other is the optical guide which is inserted into the arms of an electro-optical modulator in a Mach-Zehnder configura tion. Given that the latter will be presented in detail in a later publication [Luc 03], the description here is restricted to the essential points . This Chapter uses many of the methods detailed in three PhD theses (in French) [Ant 98, Mou 99 and Tro 01] .

I Deposition methods Deposition techniques depend essentially on the nature of the organic materials being handled. The use of vapour deposition methods for polymers is difficult, due to the necessarily high temperatures required which entail possible chain degradation, so polymers are normally processed using spin coating from the liquid state. Small molecules are easier to vaporise and are often deposited, through sublimation, under vacuum. Appendix A-II , Section II gives specific details of some optically and electronically active polymers whereas Appendix A-II , Section III presents similar, widely used small molecules.

1 Spin coating As shown in Figure VIII-I, a small quantity (several drops) of a liquid material is spread over a flat surface. The substrate spins with a predetermined acceleration, rate and period in order to control the thickne ss of the film, which is spread uniformly due to centrifugal forces . Occasionally, effects caused by the substrate edge can arise . Typical examples of films are those prepared from poly(3-octathiophene) (P30T), poly(phenylene vinylene) (PPV) and polyimides. Each material requires its own specific technique s in order to yield homogeneous films. Mostly, these material s

202 Optoelectronics of molecules and polymers

pipeu c

-----------~~

Turnt abl e rotation fnnns homogeneou s iii m

6 substrate

turntable

Figure VIII·!. Spincoating.

are turned into films using their solutions and once spread often require evaporation of the solvent and annealing of the polymer to obtain the appropriate optical properties. In the case of LEDs, this often means trying to avoid the formation of a polycrystalline material, which would otherwise reduce recombinations (see H type aggregates in Chapter VII, Section VI-6) and would lead to crystal joints limiting electron transport, and diffusion loses in guides.

2 Vapour phase deposition Vapour phase deposition consists of placing a material in a metallic pan, as shown in Figure VIH-2, which is then placed under a high tension and, with a high current, the material is heated and vaporised within the evacuated chamber. A substrate is

.. . .... . .

sample growing depos it quartz micro-bala nce _

vacuum cbamber

4 - -

evaporation cone material for evaporation evaporation pan

vacu um pump (lO " Torr)

Figure VIII-2. Equipment used for deposition by evaporation.

VIII Fabrication of optoelectronic components

203

positioned within the cone shaped pathway of vapour so as to trap condensing material. The substrate, through its support, can have its temperature adjusted with respect to the required properties of the condensed layer. Small molecules, such as Alq3 can be deposited using this method. The pans are typically made from molybdenum in two parts: a curved base to accept the material and a cover pierced with several small holes so as to multiple the number of vapour cones and limit rapid clogging of the chamber. The vacuum is generated by a secondary turbomolecular or diffusion pump, in addition to a primary pump, and is of the order of 10- 6 to 2 x 10- 6 mbar. The working temperature is controlled with a thermocouple placed in the pan. The thickness of the deposited sample can be calculated using a quartz balance . This works by altering its vibrational frequency with respect to the mass of deposited material ; normally a straight calibration curve can be prepared showing variations in frequency (Af) with respect to film thickness (measured using for example a Dektak micrometer) for a particular material. It is worth noting that metal electrodes (calcium and aluminium for OLEO cathodes and gold for electro-optical modulators) are deposited using this method , often using masks to predetermine the design of the electrodes. Generally speaking, the system often includes a series of pans so that successive depositions of organic solids can be made without exposing the formed multi-layer films to air (Figure VIlI-3).

3 Polymerisation in the vapour phase (VDP method) Vapour Deposition Polymerisation (VDP) involves either: -

-

the thermodynamic relaxation of a monomer, condensed on a cold substrate, invoking its own polymerisation (Figure VIlI-4). Examples are those of poly( para-phenylene xylilene) (PPX) and its halogenated derivatives [Jeo 91]; or the co-evaporation of different monomers, from different pans as shown in Figure VIlI-3, which, with precise temperature control of pans and carrying substrates, allow co-polymerisation to yield, for example, polyimides of the type PMDA-ODA [Brou 01].

Materials obtained using this technique often exhibit very good optical properties . In the case of PPX, a remarkable transmission has been observed in the visible spectrum [Jeo 91], which can be explained by its large gap (around 4 eV) making it an excellent candidate to encapsulate optoelectronic components functioning in the visible spectrum (electroluminescent diodes or photovoltaic cells) . The procedures described here can also be used to prepare photoguides based on polymers, for example, either by adjusting the function indices of the monomers (brominated or chlorinated PPX) or, in the case of polyimides, by doping with a colouring agent such as DR I.

204

Optoele ctronics of molecules and polymers Turntable

I'lllatioo axis

Evaporatioocone

T~

Insulator

Figure VIII-3. Representation of a mult i-pan turnt able [Tro 0 I).

4 Film growth during vapour deposition: benefits due to deposition assisted by ion beams [MiiI 89 and Smi 90] a Configuration used for deposition The titled technique combines classical Joule evaporation effects with those due to an ion beam. The result is the modification of, for example, structural, interfacia l and physical properties of the deposited film. Here we shall try and show how the ion beam works during film growth.

drawing fan

heating element vacuum gauge cooled substrate primary pump liquid nitrogen reservoir

Figure VIII-4. Polymerisation from the vapour phase (YOP).

VIII Fabrication of optoelectronic components

205

substrate

. .. .....

• 11

ion beam pan Figure VIII-S. Simplified configuration of ion beamassisteddeposition (IBAD).

b General points: nucleation and film growth a nucleation Individual atoms which hit the film surface continue to diffuse until (Figure VIIl-6) they are: -

evaporated; used up in the growth of a sufficiently large site; or captured by an existing cluster or trapped by a particular site.

The accumulation of atoms continues at a certain speed (of the order of R, = p(2nMkT)-1 /2 where p is pressure and M is molecular mass) and proceeds at a rate dependent on activation energies. ~

Ways in which thin films grow Initial film growth is conditioned by interactions between the film and the substrate , and can result in various types of growth : •

•

•

when interactions between conden sing films are stronger than those with the substrate, we obtain the growth of thick islands prior to the whole of the substrate being covered (Figure VIIl-7-a), as described by the Volmer-Weber model ; when interactions are stronger with the substrate and singularly decrease with each added layer, we obtain the formation of successive layers (Figure VIIl-7 -b), as described by the Franck-van der Merwe model ; and when a Stranski-Krastanov type growth occurs (Figure VIIl-7-c) it is due to the loss of monotonic difference s between films, and results in a mixed growth. condensation

re-evaporation

ads:;' 1! 1

a

""""0. at 'oofa" 1T1'-n-u-c-Ie-a-ti-o-n....>.LI-'-.p.,..,-~interdiffusion Figure VIII-6. Various possible mechanisms accompanying film growth.

206

Optoelectronics of moleculesand polymers

Growth:

(3) by islands

o

(b) layer by layer

0

(c) mixed

po

Figure VIII·7. Various routes to film growth.

During growth via the formation of islands, there is a second step corresponding to stable nucleation and coalescence of clusters (fusion of islands). Once beyond this step, canals can exist leaving a passageway through layers to the substrate below. Once growth is finished, diffu sion can occur across the surface, filling the canals. y Developmentofmicrostructure in thickfilms Movchan-Demchishin's diagram is usually used to repre sent the development of film structure with respect to the ratio TsjTm, in which Ts is the temperature of the substrate and Tm is the melting point of the material. Different zones can appear: •

•

•

Zone 1, where Ts jTm = 0.2 to 0.4, is a zone in which shadow effects dominate and the diffusion of adatoms (atoms added to the surface) is too weak to overcome shadows formed by lower layers. The result is a formation of a mesh (tubes) of column-like aggregates which exhibit poor adherence; Zone 2, TsjTm = 0.6 to 0.7, where we obtain columnar crystals with curved surfaces in a process dominated by diffusion effects of adatoms at the surface, and the deposited film exhibits a good adherence; and Zone 3, Ts jTm = 0.9 , where the columns recrystallise to give equiaxial cry stals in a mechanism dominated by diffu sion through volume which aims to minimise mechanical constraints.

In fact, the dense microstructure of Zone 3 obtained from adatoms with a substrate at high temperatures can also be obtained using highly energetic adatoms obtained with an assisting beam, as in the ion beam assisted deposition (IBAD) process.

c Ion effects on obtained films

a Ion - solid interactions (particularpoints in systems at low temperatures) Thermal spikes -

-

-

Origin of thermal spikes: when ions weakly penetrate the film surface, thermal spikes can result when the local temperature (energy from ion depositions) goes above the melting point of the solid . Thi s mechanism is facilitated when the cooling time (time required to dissipate energy) is longer than the time required to stop ions. Effects due to thermal spikes: energy dissipation though thermal conductivity following thermal spikes activates hops of atoms in porous zone s between adjacent sites . Modelling: for the present time, effects of thermal spikes have been modelled using classical equations of thermal conductivity. When the ratio of bombarding

VIII Fabrication of optoelectronic components

207

to incident ions (factor R) is close to I, then the growth of porous micro-columns is arrested, spikes occur and initially empty zones are closed up. This model does not plan for an increased density in zone I, as it supposes that there is a single isotropic arrangement of atoms due to the thermal spike .

The limit for the approximation of binary collisions When the velocity of particles is low (i.e. energy E < ca. 100 eV) multiple interactions are involved and the classical model, based on binary collisions and performed using the TRIM program, is no longer valid. Indeed, the assumptions that are used , that collisions are violent and only very close incident particles are involved, requires modification using a dynamic method in which co-operative displacements at the interior of the excited solid are developed. ~ Ion bombardment effects during initial nucleation and growth of thin films general, ion beam bombardment can induce :

• •

In

an increase in the size of island-like particles; and a decrease in their number density.

These two effects have been attributed, but not directly associated, to an increase in adatom mobilities. While the final result does not correspond to a set of general rules, it has been shown that the application of lEAD: • •

to a crystalline substrate increases nucleation sites; or to an amorphous substrate increases the size of islands (increasing diffusion and mobility of adatoms).

d Effect of IBAD on thin films On applying IBAD to thin films, normally, there is a modification of the film density and a collapse in zone I of Movcham's diagram . The size of the grains generally decreases as the columnar structure disappears (if the crystal size increases then there are other effects operating such as a rise in temperature). As we have seen though, the thermal spike is not representative of this mechanism of densification. Other, different models have been proposed and are detailed below.

Cascading collision model (Figure VIII-8) The titled model works by accounting for the transfer of momentum from incident ions to remote or pulverised atoms by:

(1

-

incident ions being either retro-diffused or incorporated into the film structure; and incident ions producing phonons, vacancies , electronic excitations and atoms propelled away from the surface . The latter can either leave the surface as pulverised atoms, or penetrate deeper into the film where they are trapped at interstitial sites, and in preference those sites are favourable to open zone closure.

Vacancies close to the surface produced by ion bombardment, are partially filled by atoms from the vapour phase. At a high enough R value (ration ion/vapour) the

208

Optoelectronics of molecules and polymers filling densification

densification

depleted zone

densification

r-+-,

I. I I I I I I I I I

Film

extension

(a) without ion bombardment: abrupt decrease in densification of surface, Only vapour phase deposition has effect

(bl densification following

ion bombardment without filling by incident vapour phase atoms: only bombardment effect after vapour phase deposition

(c) densification following ion bombardment with filling due to incident vapour phase atoms: ion bombardmentand vapour phase effectsare coupled

Figure VIII-S. Film densification with and without ion bombardment.

latter mechanism results in a densification from the surface to the bottom of the film, so that the film no longer grows as porous columns but as a dense material. ~

Estimations with respect to vapour phase atom and ion beam fluxes

Estimates

=

using the ratio of ion to atom fluxes (R J,/h) have been performed. With E = 100 eV , typically, a value of R = 10- 2 has been obtained [Cuo 89]. It has also been shown that the ratio can be optimised when JA/J, » Y, that is the degree of pul verisation and is fulfilled when E < I keV (where E > 1000 keV) , assuring a low degree of pul veri sation.

y 2-D simulation of molecular dynamics Th is method uses Lennard-Jones potentials to describe interacti ons between atoms on a zero temperature substrate. Figure VIII -9 shows the filling up of empty zones and the surface movement of atom s caused by a series of incident ions. Simulations have shown that the empty spaces are filled by atoms already well within the film or by atoms from the vapour pha se. 8 Example of expected modification of OLED interfaces Finally we can consider the action of IBAD on OLEDs and their behaviour [Mol 00]:

incident ion

t

Figure VIII-9. Dynamic molecular simulation of atomic rearrangement, in which atoms are pushed into the bulk, and the material collapses and becomes more dense. Dashed arrows show surface diffusion.

VIII Fabrication ofoptoelectronic components

• •

209

when used with respect to the injection of carriers (especially with respect to the anodic side as can be experimentally shown); and whenused with respectto the limitationof interfacedefaults(for exampleoxygen or non-radiative centres) allowing an increase in fluorescence yields ([).

The results obtained usingthis systemcan be interpretedusing the general effects due to a low energy ion beam during layer deposition, which are notably: •

•

an interface(ITO/ Alq3)effect. Here,ion bombardmentdirectlyeffectsnucleation during the first steps of filmformation. The result, as detailedin FiguresVIII-lO-a and VIII-IO-b, is a decreasein porosity at the interface, an increase in the specific surface contact area and a reinforcement of hole injection at the ITO anode; and a volume effect due to layer densification following collisions between incident ions and deposed atoms or molecules. The result is a limitation in the number of traps contributing to luminescence extinction and an increase in the layer resistivity (densification of zone I according to Thornton's model).

5 Comment: substrate temperature effects While there are many parameters availablewhen using ion beam assisted deposition, such as the nature of the ions used, their energies, their flux and the density of ionic current, a more classical variation can be introduced through varying the substrate temperature (see also precedingSection 4-b-y). Using this technique, a complete study was made of para-sexiphenyl film depositions [Ath 96] in which an order-disorder transition was demonstrated at 290 K. In addition, it was shown that depositions performed at a low speed but with a hot substrate resulted in the formation of crystallites (of size ca. 2I-Lm). Optimisation to obtain a good crystalline film, led to the use of a very slow deposition rate (0.01 nm sec-I) and a very high substrate temperature (200°C). Indeed, the nature of the substrate can also have an effect: crystalline substrates (silicon) tend to decrease crystallinity with respect to amorphous substrates such as glass.

'E

"! ~I .....~~~~ ...~;.::...~

e Z

(a)

( b)

Figure VIII-tO. (a) Injection without \BAD ; (b) Injection using \BAD .

210

Optoelectronics of molecules and polymers

II Fabrication methods: OLEDs and optical guides for modulator arms 1 OLED fabrication a Component structures (see Figure VIII-ll) Classically, the structure of an OLED is based on the layering of organic materials onto indium tin oxide (ITO) covered glass. The ITO serves as an anode; it is generally chosen as its workfunction is particularly well adjusted to favour the injection of holes into organic solids and is used to give a transparent layer. In its most simple and typical form, the OLED is composed of layers which serve for hole injection and transport, and electron injection and transport. The latter is in contact with the cathode and has a low workfunction; calcium is widely used along with a protecting layer of aluminium against air. Figure VIII-12 shows the necessary stages in the preparation of organic LEDs.

b Substrate preparation [Tro 01] The normal starting point is a plate of glass covered on one side with ITO, bought from a manufacturer such as Balzers or Merk display technologies who use similar fabrication techniques. The square resistance of ITO (of the order of R = 4 to 20 Q /square if possible) can be controlled by the procedure used and any treatments [Tah 98 and Wu 97] ; its transmittance is of the order of 90 %. The plates are then cut into squares using a diamond cutter, the size of which for laboratory tests is several em" . So that the same glass support can be used to ensure a firm contact with the cathode without touching the anode (see Figure VIII-13), a strip of ITO is removed using a four step chemical etching process. The four steps are: protection of the templated surface using a varnish or strips of rubber; immersion of the samples into a hydrochloric acid bath at 50 °C for several minutes; sample rinsing with water; and removal of the varnish with acetone. Cut and etched samples are then cleaned using chemicals in an especially important stage with respect to the preparation of electroluminescent diodes . ITO, in fact,

,J"I.-- - - - __. _

-'"" I

'-

I I

---'1-

aluminium

4

calcium dee tron injection and transport

--

clec Iron-hole recombination

---

hal e transport and injection

..-

anode (ITO)

substrate (glass)

Figure VIII-ll. Multi-layer diode structure.

VIII Fabrication of optoelectroniccomponents

211

Sample cutting (-\ cm 2 )

Organic layer deposition either: by spin coating: or

hy evuporation under vacuum

Cathode deposition and evaporation under vacuum (calci um + aluminium)

Figure VIII-12. Successive stages in the preparation of organic LEOs.

plays a central role in the ageing process and it is es sential to have as clean surfaces as pos sible, given that thi s can lim it the diffu sion of impurities into the organic material. The washing process consists of sever al stages in which the samples are exposed to: • • • • • •

deionised water at 60 DC and ultrasound for 5 minutes; acetone at 55 DC and ultrasound for 5 minutes; eth anol at 60 DC and ultrasound for 5 minutes; deionised water at 60 °C and ultra sound for 5 minutes; deionised rin sing; oven drying at 60 DC for I h.

Comment : it has been shown that supplementary treatment s such as plasma or ultrasound can further improve the properties of ITO .

.1 Cathode contact point without risk - ----..\ of short -circui t

'I

I ,It

ca thode organic film

_ glass subs trate

ITO

I

Figure VIII-13. Etching ITO to remove possible anod e-cathod e short circuit.

212

Optoelectronics of molecules and polymers

c LEDsfabricated for testing: supportconfiguration Figure VIII-14 detail s the layout used to place 6 electroluminescent diodes on the same ITO substrate. Emitted light traverses the ITO and the glass substrate. So called ' on top ' configurations, with the ITO deposited last onto the hole injection layer are no less intere sting than the configuration already discussed. This type of structure allows the cathode metal to be isolated from air thus limiting secondary reaction s, however, the technolo gy required to depo sit ITO is demanding and generally involves the use of double pulverisation and deposition using ' Dual Ion Beam Sputtering' (DIBS), or the use of a liquid dispersing agent.

2 Fabrication of modulator guides/arms from polymers a The importance of polymer based couplers/guides The wide range of fabrication techniques available to make polymer waveguides allows a degree of flexibility when preparing these optical conne ctors. The combination of different techniques can permit greater versatilit y in the final product, which can be manufactured reliably and cheapl y. The se optical conne ctors can be used to link optoelectronic components, to work inside specific module s or even between various module s. Optical interconnections are used in integrated optical circuits, waveguide s and space transmissions. Optical systems integrating polymer based materials are now generally considered the solution for use in connecting, cou pling and interfacing with electronic components.

Cathode Organic film

Glas substrate

ITOanodc

Figure VIII-14. Classic set up for testing 6 LEOs.

VIIl Fabrication of optoelectronic components

213

While optical fibres are used for point-to-point links between emitters and receptors, the actual topologies used require the use of passive components to distribute beams of light to the various 'clients' , a job which is fulfilled by optical couplers.

b Fabrication technology used for polymer guides Mostly, geometric guide forms are determined using lithographic techniques. A few techniques involve direct writing with a laser or a beam of electrons. There are in fact four principal techniques used in the preparation of polymer guides: a Etching Etching corresponds to the control and selective removal of material to realise the required structure. The possibility of being able to etch polymers with a high degree of precision permits the use of direct techniques to obtain waveguides . For example, a polyimide spread over a rigid substrate by spin coating can be eliminated at localised points using either an adequate reactive ionic solvent or by etching to yield a grooved channel. ~ External diffusion A large number of polymers are well suited to receiving doping agents (even selectively). However, certain materials trapped within the polymer, particularly those with low molecule weights, can be thermally diffused to the exterior of the material (polycarbonates are a good example). Zones treated using these properties can give rise to optical guides.

y Polarisation and localised reactions We can selectively create zones corresponding to optical guides by using molecular doping agents polarised in the same direction. Localised oxidation reactions can lead to bridging reactions within a guide zone . I) Induced diffusion and polymerisation Guides can also be obtained by using the controlled diffusion of monomers within a polymer matrix. The above techniques exploit properties particular to polymers using specially designed apparatus. We shall limit ourselves to detailing the etching process, which is the most widely used of techniques.

c Guide fabrication by etching Figure VIII-IS shows the two principal steps required in etching. Firstly, to define the geometrical form of the optical component, masks are prepared through the use of a photoresist which is itself made of photosensitive resins i.e. organic materials, and then secondly the organic film is etched (or in some cases photo-polymerised) using the mask . a Photosensitive resins for mask fabrication Generally, resin based polymers are used in the preparation ofetching masks, which are subsequently used to etch thin films formed from insulating, metallic or semi-conducting materials. Two types of resin are available: positive resins in which the treated part becomes soluble (developable) in a selective solvent; and negative resins which once treated become insoluble.

214

Optoelectronics of molecules andpolymers For "high energy" lithography. AI film thickness is I urn For RIE. RlBE and lBAE. AI film thickness is 400 nm

rr-'l....

-

...

Mask

Photosensitive resin 1813 Al deposit is 4000 A (or I urn) Adhere nce promoter VM 651 + PI 2566 ~~~~======~r- Si02 5000 A Si

UV irradiation

Photosensitive resin Shippley 1813 AJ deposit is 4000 Adherence promoter VM 65 I + PI 2566 Si02 5000 A Si

Resin development 5!Jm

..........

Photosensitive resin Shippley 1813 Al deposit is 4000 A Adherence promoter VM 651 + PI 2566 Si02 5000 A

...

~

Si

Attack at aluminium Figure VIII-IS. Principle used forpolymer based lithography (example is polyimide PI 2566, Du Pont).

The form of the mask is defined by a tracing UV light over its surface. Subsequent development using a wet process results in perpendicular walls (negligible 'underetching'), if the UV is used in perfect parallel beams. Different developing liquids can be used, such as acids (HF, Hel, H2S04, HN03, H3P04) , bases (KOH, NaOH, NH40H) and oxidising agents (H202, N2H4)' These resins must, in addition, be able to resist the wet or dry processes used in the following second step. If this is not the case, then a supplementary step using aluminium is required (see Figure VIII-IS). ~

Etching of optical structures (polymer guides) Wet etching can be used in the preparation of polymer guides, however, there is the non-negligible problem of

VIII Fabrication of optoelectronic components

215

' under-etching'. Wet development proceeds in an isotropic manner and can attack supporting zones around the mask, resulting in non-rectilinear profiles altering the geometrical definition of the guide . At present, the tendency is towards dry procedures which use gases in the form of plasmas .As the size of the details required decreases towards l urn , design duplication at such high resolutions is no longer possible without anisotropic etching. This can be achieved using a masked ion beam which, perpendicular to the target , results in film pulverisation. In addition, in terms of economics, safety and environmental hazard , wet chemical processes are onerous and necessitate strict security controls. The final thickness of the guide is equal to the initial thickness of the film. Typically the guide section is rectangular or cubic in shape . The strips present an interface with air and exhibit high refractive indexes and typically function as multimodal guides . Such channels can eventually be filled by a second polymer with a different indice. We can thus obtain embedded guides which can function in mono- or multi -mode . Simultaneously, the second polymer can act as a protective layer against external shocks. Guides fabricated in such a way exhibit uniform profiles and indices, resulting in well-defined properties. In order to limit losses during use, a great deal of care is required to arrive at smooth etched walls with definition greater than 0.1 IJ.,m. Dry etching can be, in practical terms : • • •

completely 'physical ' where ionic etching with inert gases, such as argon ions, are used to physically pulverise the material surface ; a combination of 'physical ' and chemical etching, for example the use of reactive CF4 ions to etch Si surfaces ; or purely chemical, for example the use of oxygen plasma to remove, by oxidation, a photosensitive resin resulting in the production of volatile gases which can be readily evacuated.

Appendix A-9 gives further details. The etching mechanisms depend both upon the nature of the gas used in the plasma (which gives rise to reactive ions or species) and the technology employed, which uses one of three principal configurations. • • •

plasma etching or reactive ion etching (RIE) where substrates are placed on the lower electrode placed with respect to a radio frequency plasma; reactive ion beam etching (RIBE) where a beam of reactive ions, for example 0 +, are accelerated towards the substrate from a source ; and ion beam assisted etching (IBAE) which couples the effects of an incident ion beam (which for example can be inert gas ions such as Ar" or reagents such as 0 +) with the effects of a flow of reactive gas such as molecular oxygen.

Figure Vlll-16 schematically summari ses the different techniques which can be brought into play in various etching processes. In all these proce sses, the etching reactions are initiated by ions which are accelerated perpendicular to the film surface . Consequently, these proces ses can result in high contrasts with practically vertical walls and examples include that of polyim ide films

216 Optoelectronics of molecules and polymers

E(eV )

> 10 k.\,

100 Polcnllal en "10 of cheml lIy adh'. Ion

10

I...,.. nu, hUI h l~hl , u n '~l rup'('

Ch. mlCIII pol nllal en "10 of rad lca ~ a nd m ta -cta bl fo"," bOlropic nu x

Chcmlcal IMllrnlialrlll'flO

lliahl ..... ctb e and directed nu

Rlt:

moelh. Ion .Ichlna

HIE 'en vlronme m' l~ of a dtreeted nu, of Ion unde r innu nee of several hund rrd .\', hoI It'dmn\ a nd an ('DImple n o,",' of chrmJeall) acli ,r \p«h.""i

1II,\ t: I, .I mllar 10 RIE hUI d 001 u hot eleelro ns, U ~ rc", or no nldical or meta -st hi. fonns In Ih reage nt flu\.. TIll' chcmkal poten tlal en "10' correspond In a tem pe... tu re do~ In ambit o. and the reac tlve nux I both direc ted an d in a ronlrolled ch("mJcal state, tlu.II " mA t: exhibit> a red uced l....lrop ' wlth re pect 10 RIE.

' .' h,wCIII' etching I, RI8E \1." '" onl, a . Inlll' orion!li ""hich is both .........Ilall' Ih rou ~h lb. pun-I) reaethe and \1.'" of ch.mlcall, Inr rt dlreeted hUI of 10 w lon.' "hlrh resu lts In a hanJ la) er hut ".urr('n Int cn~tl from 10" ' seIrrth 'll l

no,",

Figure VIII -16. Comparison of techniques in the etching process (RIE, IBAE, RIBE, and non-reactive dry etching).

which have been totally converted. In addition, these dry processes are considerably more reproducible than wet processes, and because of the wide variety of available parameters (reactor pressure, gas composition and flow speed, radio frequency power, energy and current density of ion beam), control can be exerted over characteristics such as etching speed, etching selectivity with respect to materials used, and the lateral wall angles of etched features . While these technologies were initially developed for inorganic electronic/ optoelectronic applications (etching Si with CF4) they are also well applicable to polymers. Indeed , the latter can be actually quite resistant to such processes, due to the strong backbone links which make up the polymer chains (particularly so for polymers containing aromatic or polar groups), but the use of these materials with aluminium masks and reactive beams containing oxygen, which reac ts with carbon based chains, can yield remarkable results . Figure VIIl-17 shows an example [Cor 01]. Transmission losses , estimated using the 'cut-back' method , are around 0.5 to

VIII Fabrication of optoelectronic components

217

Figure VIII-I7. Polyimide guideobtainedusingRIBEandcleaving(0+ ion beam withenergy 2.5 keV). 4dB cm- I in TE mode where TM has}. = l.3[Lm and}, = 1.55[Lm . (This destructive method measures the transmission of a guide through successive and progressive reduction in wavelength used; Section IV-2 has further details) .

III Photometric characterisation of organic LEOs (OLEOs or PLEOs) Photometry is the term used to describe the characterisation of rays from system containing luminescent units [Des 91) . From a study of the energy of the rays, photometry can yield an evaluation of luminosity, which can also be described in visual terms when the detector is the human eye. The terminology used takes account of this binary aspect by using in the notation the indice 'e ' for energetic terms and 'v' for visual terms .

1 General definitions a Energy and luminescence fluxes In general, flux characterises the rate of flow of rays: -

Energetic flux (radiant) e emitted by a light source is quotient of the quantity of energy dW e transmitted during time dt at an interval dt. We can therefore write this as:

aw,

e = -

(1) dt If the flux is uniform over time t then e = W elt and is expressed in watts (W) .

218

-

Optoelectronics of molecules and polymers

The luminous flux Q>v is defined with respect to the sensitivity of the human eye to different wavelengths of light. The value is expressedin lumen (lm).

To convert from energetic to luminous scales the constant Km = 683lm W- I is multiplied by the diurnal photo-response V(A) of the human eye. The function V(A) is zero outside of the visible spectrum, which is between Al = 380nm and A2 = 780 nm. The diurnal photo-response reaches its maximum at around 555 nrn, as shown in Figure VIII-I8, in which [V(A)]Max = 1 at A ~ 555nm . Night vision, however, results in the same maximum moving to A ~ 510 nm.

a Relationship between energetic and luminous flux for a monochromatic or pseudo monochromatic source Given the above indicated variation required when going from energeticto luminous (visual) scales, for a monochromatic source with a given emission wavelength Act, the luminous flux (v) can be written as a function of the energetic flux (e):

(2) Classically, eqn (2) is also used for a polychromatic source with a maximum emission centred around the given wavelength Act ; eqn (2) can therefore also be applied to a pseudo monochromatic source. ~

Relationship between energetic and luminous flux for a polychromatic source

More rigorously speaking, the flux contributed over all energies E (= hv = hC/A) should be taken into account when consideringa polychromatic source. The spectral distribution on a photometric scale, here the flux (<1» , is characterisedby the function <1>' (A) in which <1>' is the spectral scale definedby: '(A) = lim (&<1» 8A--+O

IIA

where &<1> is the fraction of flux contained in a spectral band of size &A about the wavelength A. Knowing <1>' (A), we can now calculate the total flux (A. a, Ab) within

YeA) 0.8

0.6

0.4

0.2 0 350

400

450

500

550

600

650

700

750

Wavelength (nm) Figure VIII-IS. Photo-optical response normalised with respect to the human eye: Y(1)

VIII Fabrication of optoelectronic components

219

a spectral band [)"a, Ab): (3)

Once we know the total energetic flux (e). which as detailed later on is measured by a photodiode, then when studying for example diode electroluminescence, the normalised electroluminescent spectra (I:) needs to be measured to find the spectral energetic flux ( ~CA.)) . It should be noted that: -

10

00

SeA) . d x, corresponding to the sum over the area of an empirical I: = normalised spectra; and e = aI:, in which 'a' is a constant.

'(}.)

We can go on to deduce that ~(A ) = a S(A) , that is: a = Ee = S(}') . On knowing the electroluminescence spectra, we can thus go from the energetic flux e to the spectral energetic flux ~ (A) as in: <1>' (A) _ e . SeA) _ ----::-::,e_ .- · _S_(A_)_ e I: - 00 SeA) . d),

10

(4)

For a given wavelength A, the general method used to convert from an energetic scale to a luminous (visual) scale once applied to a spectral scale permits, for the latter, the eqn (5) to be written, in a form which resembles that of eqn (2): ~ ( A ) =

x., V(A) ~(A) .

(5)

On applying eqn (3), the luminous flux is given in the form: v = in which

~ (A.)

f ~ (A.)dA. = f V(A.)~(A.)dA., x.,

(6)

is given by eqn (4).

b Energetic and luminous intensities: definition of the candela

Energetic intensity (radiant) The energetic intensity (Ie) is defined as the ratio between the emitted energetic flux (d
Ci

(7)

If the emitted flux within a given solid angle (Q) is constant, then Ie = intensity is thus expressed as watt per steradian (W sr- I ) . ~

%; the radiant

Luminous intensity The luminous intensity (L) is defined by the equation I , = ~~v ; when v is a constant within Q, I, = ~v . I, is expressed in lumen per steradian (1 m sr- I ), otherwise denoted the candela (cd).

220

Optoelectronics of molecules and polymers

The International Commission on Illumination (CIE for Commission Internationale de I' Eclairage) defined the candela as a luminous intensity, in a given direction, for a source which emits a monochromatic beam with frequency 5.40 x 10 14 Hz (corresponding to wavelength 555 nm) and energetic intensity in that direction of 6~3 W sr- I • This definition takes into account the constant Km (683 1m W- 1) which was introduced above. For a monochromatic (A = Ad, or pseudo-chromatic ray

(8) For a polychromatic source, however, the reasoning we applied to tie v to ~ can be used again to relate I, and I~ . This gives a relationship between the latter two quantities an equation which resembles eqn (6) for v and ~ :

r, =

Km

f V(A)I~(A) dA,

(9)

where I~ is given by an analogous relationship to eqn (4), i.e.

e_ I' (A) = _I · S_(A_)

~

e

= 10

Ie . SeA) SeA) . dA.

00

(10)

c Energetic and luminous luminance ex Energetic luminance The energetic luminance (Le ) of a source is defined as the ratio of the emitted energetic intensity to the area of apparent emitting surface (Sa):

r,

I dc

t, = Sa = Sa dQ ; for a constant flux, Le

(II)

t, e, = -Sa = . QS

(12)

a

Energetic luminance is expres sed in W sr- I m -2 and is the flux emitted in a unit solid angle per unit of apparent source surface .

Comment: the physical significance of luminan ce If As is the surface area of a source, the apparent surface is Sa = As cos ex, as detailed in Figure VIII-19 . The luminance is the direction 00' is the intensity per unit apparent surface and characterises the aspect observed by an observer at 00'. By defining the light incident on a screen from a luminous point on a surface A~, with the help of the relationship E = ~~ = = i, s we can conclude that 2 s s sources of the same intensity result at the same screen in the same effective lighting . These two sources can come from different surfaces , for example, the large surface such as that provided by a fluorescent tube or a small surface due to say a lamp filament. And the fluorescent tube will appear to be less bright than the filament and it is this characteristic which is placed in terms of source luminance.

trf,

f"

VIII Fabrication of optoelectronic component s

221

FigureVIII·19. Surface apparent to a screen. ~ Expression for visual luminance defined by

The visual equivalence (Lc) of luminance is

t,

when the flux is constant

I d v

L - - - - -. v - Sa - Sa dQ '

(13)

t, v Lv = - = - Sa QS a

(14)

which is expressed in cd m- 2 . For a monochromaticor pseudo-monochromatic source,the visual luminancecan be determined, if we know the (constant) energetic flux of a source, using eqns (14) and (2) to give: v x, V(}"d)e Lv = = (15) QS a

QS a

For a polychromatic source, there is the relationship between L, and again resembles eqn (6) for v and ~ and which is:

t., = Km

f V(A)L~(A)dA

L~

which

(16)

where L~ is given again by a relationship analogous to eqn (4) i.e. L' (A) _ t, . SeA) _ _L--'..e_·_S(_A_)_ 00 e ~ SeA) . d), .

10

(17)

2 Internal and external fluxes and quantum yields: emissions inside and outside of components a Restrictions limiting the passage of the internal ray to exterior of an OLED FigureVIII-20detailsan emissionfrom an OLEDtowardstheexterior. The rayemitted by the emitting layer reaches the organic phase-air interfaceat an angle of incidence El which can be greater or less than the critical angle Ell defined by the relationship nsin Ell = nair = I, in which n is the refractiveindice of the organic material.When El > Ell, the flux is transmittedto the exterior,however, when El > Ell , the fluxundergoesa total internal reflection and is, a priori, returned back into the OLED structure without reaching the exterior.

222

Optoelectronics of molecules and polymers

Preliminary comments

a neglected effec ts When n = 2, Ell ;:::: 30 0 , which being quite a small incident angle and not far from the normal incidence means that we can use the reflection coefficient in the form R ;:::: (~~:) 2, which yields a coefficient of transmission of T = 1 - R ;:::: 75 %. However, if n = 1.5, we have T ;:::: 96 %, although Ell ;:::: 40 0 2 which renders debatable the use of the equation R ;:::: ( ~~ :) . Given the relatively low indices that organic materials exhibit, T is generally accepted without correction and is approximated to T ;:::: 100 % = 1 when El < El,. In addition, the organic, ITO and glass layers are assumed to be uniform and the indice under consideration remains that of the organic materiel and air only. ~ Effects due to limitin g angle As shown in Figure VIII-20, the limiting angle imposes the restriction upon the emitting rays so that only that with angle El < Ell can leave the structure, so:

•

at the OLED interior, in the recombination zone, emissions going towards the frontal surface occur in a half space, that is QI =

•

f

S

dQ 1 = 271:

r =rr/2sin odEl = 271: ;

J s=o

the portion of flux which can leave the OLED is emitted from the emission zone within a solid angle limited to Q2 = 271:

l

S= SI

sin El dEl

= 271: (1 - cos Ell).

0= 0

Given that sin Ell so that we have

= I In, the term C can be calculated by through C = [1

- cos Elil

Rc,"f,.'omhin;Jlion l one

leadinf: to emiwion al surface SD = ~Sll

Figure VIII-20. Schematisation showing how only certain emitted rays are allowed to esca pe to exterior due to critical solid angle Q2 of the surface angle 81 (Q2 = 2lt[ I - cos 81D.

VIII Fabrication of optoelectronic components

223

finally we obtain Q2 = .; from which can be deduced that: n QI Q2 = 2n 2 '

(18)

From eqn (18), it is evident that the solid angle (Q2) which is used is only a fraction of the total emi ssion angle (Q I) and that external yields are con siderably effected. In effect , the exteroal flux is only a fraction of that emitted. Additional comment: If we also bring into play emissions toward s the back of the device, that is toward s the cathode, then Q 1 can be obtained by taking an integral e equal to [-1£/2 to 1£/2]. On assuming that the cathode is a perfect reflector, then Q2 is obtained by integrating over e from [- el to ell. In both cases the the result is simply multiplied by 2, however, the relationship between QI and Q2 in eqn (18) remains unchanged.

b Isotropic internal and external emissions according to Lambert's law [Gre 94] Here we return to looking at the emitting material/air interface but with a ray incident to this interface with e < el. Refraction laws modify the geometry of the solid angle cone s of the emis sion , as shown in Figure VIII-21. We now have nsin e = sin

cos

The elementary solid angle s dQ 2 and Figure VIII-21 are such that cos
sin e de

I

sin

n J (n 2 - sin 2
---=

When n ~ 2 and sin 2

Recombination zone

or Indict n

leading Co emi: ion t

urf""" So =

JdSo 1....1 internal n:Ocction 8>

e.

Figure VIII-21. Internal and external emissions of an OLEO

224

Optoelectronics of molecules andpolymers

c Determination of internal and external emitted fluxes We shall limit ourselves in this subsection to calculating only fluxes emitted towards the front of the device structure. As indicated in the end-comment of the preceding subsection, to take into account emissions towards the rear of the device, the results need only to be multiplied by 2, assuming that the metallic cathode presents a perfect reflecting surface. All ratios of emitted fluxes and thus yields remain otherwise unchanged.

a Total internal flux emitted inside the structure The luminous intensity directly emitted by the source is denoted as 10int . Assuming the material to be homogeneous and having no internal interface, the internal emission from the recombination zone is isotropic and the total emitted flux inside the forward half space can be written in the form: Tint

=

I

lOint dQ

= 2JtlOint

1/2space

l

rrp

= 2JtlOint.

sin e de

(19)

0

~ Fraction of total internal flux emitted towards the exterior The emission with luminous intensity 10int remains isotropic in nature, however, the emission angle is limited to el, as we saw in Section a. By denoting ie as the flux from the interior towards the exterior, we now have:

ie = 2Jt 10int

61

1 o

sine de

= 2JtlOin(l

-

COSOI)

1

1

= 2JtIOin-z = Jt10in2 2n n

(20)

y Emitted exterior flux If luminous intensity emitted with respect to the normal to the surface of the LED is denoted by IOext , then as detailed above, the emission follows Lambert's law (lext = 10ex cos exd emitted over the forward half space is thus in the form:

ext =

I

IOext cos
ext = 2Jtloext

1/2 space

l

rrp

cos

0

= 2JtlOext 10t'" sin
(21 )

8 Relationships between fluxe s Given that the emitted external flux is a fraction of the total internal emitted flux, and conserving the overall beam power, ie = ext. and thus JtlOin ~ = JtIOext. from which can be deduced that lOin = IOextn2 . Placing n this expression into eqn (19) gives:

z

Tint = 2Jtn IO ext or ext =

1

- 2 Tint .

2n

(22)

VIII Fabrication of optoelectronic components

225

We can obtain the same result directly by the division of eqns (20)/(19) as ie = ext . The factor 1/2n 2 represents the actual yield or 'optical yield' of rays emitted from the LED. d External and internal quantum yields

a External quantum yield The external quantum yield (l1exd is defined as the ratio of the number of protons emitted by a diode (NPhext) in to the external half space over a time t divided by the number of electrons injected (Nel) over the same period of time. Ilext can be expressed therefore in the relationship: Ilext

NPhext t Nphext =-- = -. t

Nel

Net

(23)

If Ie represents the current injected into an electroluminescent structure, we have: Ie = Q/t = Nelq/t, where q is the value of elementary charge (q = 1.6 x 10- 19 C). Thus: Net = Ieq (24) t

For a given wavelength A.d' the energy of the photons is determined by Eph = hC/A.d . With the emitted external flux being ext = Welt (eqn (I) for constant flux over time t), now Nphext

Nphext A.d = -We , that IS. - = -ext. Eph

t

he

(25)

From eqns (24) and (25), eqn (23) for 11ext can be written, for a pseudomonochromatic ray, as qt..d ext (26) Ilext = hc

T '

~ Internal quantum yield The internal quantum yield is defined in the same way as the external quantum yield, with the exception that the number of photons emitted internally in the half space (Nphint) is used:

. _ Nphint t _ Nphint 11lOt - - - - - - - . t Net Nel With Np~inl = ~~ int (in an expression similar to eqn (25), but for internally emitted photons), we have qA.d
T'

Given eqn (22), on making the division eqns (26)/(27), we directly obtain: I 11ext = 2n2 Ilint -

(28)

226

Optoelectronics of moleculesand polymers

3 Measuring luminance and yields with a photodiode a General notes on photodetectors: sensitivity and spectral sensitivity Typically, a photodetector, or photodiode, gives to a responsive current (Ipho) in a form arising from two origins: •

•

dark current (10) which appears once the photodiode is placed under darkness while polarised in the same way as when exposed to light. This current 10 can arise from two causes. The first is of thermal origin and is associated with the liberation of conducting electrons by thermal excitation, and the second is associated with the generation of charge carriers by ambient beam effects (natural rays); and photoelectric current (lp) associated with the formation of photo-carriers, such as particularly mobile photo-electrons.

ex Sensitivity For a given photodetector subject to a lum inous flux, consisting not necessarily of mono-chromatic light , there is a resulting current: Ipho = 10

+ Ip.

Normally, if conditions remain unchanged during the analysis then the current 10 remains constant, however, I p will change proportionally with the received flux (cf>e), as long as the photodetectors acts linearly. So: Ipho = 10

+ crcf>e

And this expression is only valid for fluxes which are such that cf>e < cf>s, cf>s being the level above which the photodetector is saturated, as detailed in Figure VIII-22. In general terms, the sensitivity of a photo detector is defined by its characteristic slope shown in the above Figure, i.e. c = ~I:: . For a non-linear photodetector, the value is dependent on the incident flux. However, for a linear photodetector, cr is a constant, up to cf>e = cf>s, and takes on the form : (29)

slope o

10

o

ll>s

Figure VIII-22. Linear photodiodecharacteristics.

VIII Fabrication of optoelectronic components

227

~

Spectral sensitivity The spectral sensitivity, for a particular ray of light of wavelength A, is defined by the relationship (30)

For a photodetector exposed to a monochromatic beam of a given wavelength Ad and flux e(Ad), the total number of photons received by the detector per second (np) is np = Ad
= l]Tnp = l]T

Ad e(Ad) hc

= G(Ad).

This coefficient, just like the intensity Ipho of the resulting current, is dependent on the wavelength of the incident light. For a linear detector, the component Ip of Ipho is in the form Ip = FqG where F is a factor of amplification. On taking into account

we arrive at

which can also be written cr(Ad) = Ip(Ad) . e(Ad)

(30')

If the light is polychromatic, the current lp is therefore such that :

Ip =

f

dIph (~)

f

cr(}')de(}.)

f

= cr(A)~(A)dA =

f I~(}')d}'.

(31)

We thus have:

r,

-e -

f cr(}')~(}')d}' e

(4)

-

f cr(}')S(}')d}' f S(A)dA

-cr . - av-

(31')

and the value cr av then obtained for the ratio Ip/e appears as an effective (average) sensitivity. Furthermore, eqn (31') also gives: _ I ---;;-,,-f_S(_A_)d_A_ e - pf cr(}')S(A)dA . With eqn (4) we reach: ' ().) _ I SeA) e - P f cr(A)S(A)dA .

(32)

228

Optoelectronics of molecules and polymers

In addition, eqn (30) gives ~ (A) = I~(A)/a(A), which leads, by comparison with the previous eqn, to , a(A)S(A) (32') Ip(A) = Ip a(A)S(A) dA

J

.

b Example photodiode set-up [Ant 98] Photometric measurements are typically performed with a photodiode that has as large a surface area as possible, for example of the order of 100 mm 2 . a Example values for a(A) As an example, Table VIII .1 shows values given by Radiospares for a standard diode, with spectral response between 350 and 1150 nm, for a(A) at different given wavelengths A = Ad. ~ Device set-up In order to measure the luminance of OLED components, the photodiode is placed up against a window open to the OLED and is connected to an amplifier. Thus by the action of a resistor (Rc) placed in the circuit detailed in Figure VIII-23, the assembly permits determination of the value of I p from the voltage (Vp = RcIp) indicated by a multi meter. (Technical note : the amplifier typically used for this sort of assembly is powered by a tension (Va) equal to 18 V, while the tension applied at the photodiode (V phol is of the order of 10 V. The potentiometer, tied to an operational amplifier, allows this offset tension to be disregarded. The charge resistance is fixed at 995 ohms).

Y A classic set-up for measuring luminan ce: representation ofn and apparent OLED surface (Sa) In general terms, Sa is the apparent emitting diode surface, and is such that Sa = As cos a (see Figure VIII-19) . Practically speaking, the measurement of a beam emitted by an OLED is effected by placing a photodiode at the window of the measuring cell, as shown above in Figure VIII-24. The surface of the photodiode is perpendicular to the beam direction, so if we designate the surface of the OLED by SD, the apparent surface Sa is such that Sa = SD. Typically, the surface of the photodiode (SPh) is around 1 cm 2 while the surface of the photodiode is more like several mrrr' . In effect, the source acts as a point with respect to the receptor.

Table VIII.I. Principal sensitivity values for a standard photodiode ).,(nm) 400 450 500 550

a( ).,) (A W- 1 )

)., (nm)

a( ).,) (A W- 1 )

0.12 0.2 0.25 0.28

600 650 700 750

0.32 0.35 0.38 0.42

VIII Fabrication of optoelectronic components

229

Photodiode

-Vpbo

FigureVIII-23.Amplified photodiode set-up .

The half angle at the apex ( = r /D. Given typical values, that is with r ~ 5 mm and D ~ 40 mm,

Source (So)

Photodiodc (sp/»

FigureVIII-24. Simplified scheme of diode/photodiode arrangement.

230 Optoelectronics of molecules andpolymers

c LED luminance determination using the present example of an OLED a A mono- or pseudo-monochromatic LED (A = Ad) In practical terms, if Ip = Ip(Ad) is the current generated by a photodiode under a beam of light emitted by an LED at a solid angle Qph, then given eqn (30') , e(Ad) is the flux ph ' and we can write: <J(Ad) = Ip/ph. Ip is such that Vp = RcIp (see preceding Section ~). From eqn (12), (33)

Introducing eqn (33) directly into eqn (15) directly gives: Km V(Ad)V p . RcQphSO<J(Ad)

Ly =

(33')

We can also write (34) It is eqn (34) which is widely applied, even to polychromatic sources, however, as we shall see in such cases, the factor D should be replaced by a factor called 'A' which will be defined below. Clearly, the ratio A/D will indicate the extent to which the approximation of a polychromatic to a monochromatic source remains correct. ~ Polychromatic source

Following eqn (30), we have <J(A) = I~(A) /~(A) ; from eqn (12) we can write, L~ = ~ /QphSO and <J('A.) = I~('A.) /QphSoL~('A.) . On deducing that L~ = to give:

Qp~SD ~~i

'we can go on to substitute the value ofI~('A.) given by eqn (32'), ,

Ip SeA) 00 QphSo f o <J(A)S(A)

Le = -

ax

(35)

•

On transferring the value of Ip = Vp/R c in to eqn (35), we obtain

L~ (A)

=

Vp RcQphSO

00

SeA)

fo <J(A)S (A) . ax

.

(36)

The desired expression is finally obtained by introducing eqn (36) into eqn (16): VpKm fV(A)S(A)dA t, = RcQphSO f <J('A.)S(A) dA'

(37)

which can also be written as VpKm t,

.

= ARcQphSO' with A =

f V('A.)S('A.) d'A. f <J(A)S(A) dA =

Vay <Jay ·

(38)

(<Jay is defined by <Jay = f <J(A)S(A) dA/ f SeA) dA and Vay in a same way by Vay = f V(A)S(A) dA/ f SeA) dA).

VIII Fabrication of optoelectronic components

231

Table VIII.2. Comparison of A and D terms for green and blue diodes.

Diode Green (532 nm) Blue (467 nm)

A (lm A- 1) 1485 698

2224 256

0.67 2.92

y Numerical application [TroOJ} By way of example, and for two differentOLEOs, Table VIII.2lists values for 0 (pseudo-monochromatic) and A (polychromatic) terms. The latter is calculated by numerical integration of empirical values obtained over the electroluminescent spectrum. One of the OLEOs emits light centred on the green light (Ad = 532 nm), while the other is centred on blue light (around Ad = 467 nm). We can see that the coefficients A and 0 are neither equal nor proportional. The use of the pseudo-monochromaticapproximation results in: • •

overestimation of polychromatic luminancefor an OLEO which emits in the green region as OOLEDgreen > AOLEDgreen; and underestimation of polychromatic luminance for an OLEO which emits in the blue region as OOLEDblue < AOLEDblue.

We can thus see the interest in determining luminance through a polychromatic 'configuration' which does not penalise blue OLEOs, that in principle exhibits lower luminances than green OLEOs. It is worth noting though that the precision available for polychromatic observations is tied to that of the experiment used to determine the electroluminescent spectrum. d Characterising yields

a External yield The external yield is given by eqn (26), that is Ilext = h~d CI . ext has already been calculated using eqn (21), and ext = n IOext . With the luminous

laser systemwith constant opening

Figure VIII·2S. Schematisation of the equipment used to measure losses due to propagation.

232

Optoelectronics of molecules and polymers

intensity given by eqn (7), which represents the flux within a solid angle unit, here
=

In addition, lext IOext cos lp, and with I Ip F' 11 '" _ I - 1l Qph a(I..)· ma y, 'i-'ext - Jt Oext - Q ph

=

lp ~

=

=

7° and cos lp 0.992 ~ 1, lext IOex d '" - 1l Vp Placi a(l..) , an 'i-'ext - Qph Rca(I..)· acmg thiIS into Ip

eqn (26) yields:

Vp he QphRe a()..) t, q

Jt

)..

Ilext = - - - - - - .

(39)

~ Internal yield According to eqn (28) , we have llint = 2n211ext. which allows us to write (40)

IV Characterisation of polymer based linear wave guides [Col 98 and Ebe 93] The titled characterisation is generally performed using wavelengths used in telecommunications, that is 1.3 and 1.55 11mand allows analyses of: • • •

light transversally diffused by the guide; losses due to propagation, injection and diffusion at the extreme surfaces; and the transversal profile of the beam issued from the guide.

1 Measuring transversally diffused light Figure VIII-23 schematises the equipment used to specifically mea sure losse s due to propagation within the guide, a loss especially important when using polymer based materials. The source used in the system is a fibre based laser diode, which emits a transversally sinusoidal beam. A collimator, placed at the exit of the fibre, gives rise to a plan limited wave with a divergence minimised by diffraction. A 20x multiplying lens 'injects' the wave into the guide (G) and the position of the lens and the guide are optimised with the aid of micro-positioning elements. Light diffused by the guide

· x'

guide

z -=========~=========-. . z' x Figure VIII·26. Configuration for analysis of transversal.

VIII Fabrication of optoelectronic components

233

o Guide

- --I••

P

Figure VIII-27. Configuration used to measure losses by 'cut-back' .

and perpendicular to propagation direction zz' is analysed using a camera equipped with a vidicon tube (sensitive in the range 400 to 1800 nm). The video signal, resulting from a sweep parallel to xx' is digitised and treated with a computer program (Figure VIIl-26) .

2 Loss analyses using 'Cut - Back' and 'Endface Coupling' methods a 'Cut - Back' Initially, a collimated laser wave is injected into the guide using a 20 x lens and global losses are determined by observing the difference in power between the injected (Pe) and exit (Ps) light, as detailed in Figure VIIl-27. Following the simple principle method of 'cut- back' , measurements are repeated using different lengths of guide (z.), yielding a curve shown in the upper part of Figure VIII-29 with a slope due to propagation losses .

b 'Endface Coupling' [Ebe 93] The method of 'endface coupling' uses the set-up shown in Figure VIIl-28 and allows differentiation of losses due to injection from other losses . A light beam traverses the guide from edge I to edge 2 initially in order to select the modal profile exactly adapted to the guide . An autocollimator made from a lOx lens and a mirror M injects across

guide

lOX

M 'c hopper'

.-

Figure VIII-28. Layout for 'endface coupling ' equipment.

234

Optoelectronics of molecules and polymers

dB Cut- Back :

10

Lea= Lp z + Lc with

Slope: losses by propagation

OEC :

(L~

..-5-

······································X· ··············

.

--------------------1--------------------------._.J~~._._._. "_,_,_,,_,_,_,,_,_,,_,_,, nappropriate modes

LMM

FRESNEL

LoEC = Lp Z + L~

with

L~ =LFR + Ld

Diffusion (edge) Ld

o

z (guide length)

Figure VIII-29. Curves permitting analysis of different optical losses (see text for definition of parameters).

side 2 the initially filtered mode without any losses due to interference. With several guide lengths, we can obtain the curve denoted by letters OEC in Figure VIII-29 .

c Curves representing optical losses Figure VIII-29 shows various curves which allow analysis of the relative weight of different losses : • • • •

• •

LCB losses measured by 'Cut - Back'; LOEC losses measured by 'Optimum Endface Coupling' ; Lp losses due to propagation; LpR losses due to FRESNAL (reflection) ; Ld losses due to diffusion at terminal edges; and LMM losses by injection due to imperfect covering of injected and modal profiles.

IX Organic structures and materials in optoelectronic emitters: applications and display technologies

I Introduction During the last half of the 20th century there has been an extraordinary development in communication and information technologies. The dominant tool used for visualisation in televisions, computers, radar screens, medical imaging, to give j ust a few examples, has been the cathode ray tube (CRT). Recent requirements, for portable computers, phones, watches and so on, which require lightweight , unwieldy screens operating at low voltages, have also vitalised the development of new electrical component s such as light emitting diodes (LEDs) and, less recently, screens based on liquid crystals. Given the rapid pace of development in display technologies and its results (micro-point plasma screens, electroluminescen t screens based on inorganic and subsequently organic materials) it is now possible to imagi ne performances which were previously inconceivable in devices such as wide-screen (greater than 35/1 ) high definition televisions, light ultra-thin screens which could be easy to move or store, and flexible screens permitting ergonomic designs in cars and civil and military aircraft (dashboards, screens in helmets for weapons sighting and so on). In this Chapter, we have chosen to present the main types of display systems, and then to compare organic component technology against that of classic components. We shall also rapidly detail the process of converting an incomin g electrical signal into visual inform ation in optoelectronic components. This process is typicall y due to luminescence (luminous emissions not entirely thermal ) which can be provoked via various route s, such as photonic excitation (generally in the ultraviolet), excitation caused by excited particl es (cathodic lumin escenc e due to electrons), and application of an electrical field (electroluminesce nce) to give the principal examples [cur 56].

II How CRTs work CRTs operate on the principle of cathodoluminescence. This is due to the impac t of a beam of accelerated electrons on a surface dotted with ' phosphors' points, or contacts, which on relaxation to an equilibrium state following excitation, emit light in the

236

Optoelectronics of molecules and polymers

Figure IX-I. Arrangement of 'phosphorus' blue (b), green (g) and red (r) contacts on a cathode screen.

visible region . Sections VII-I and VII-2 detail the process of electroluminescence in inorganic materials and, in particular, these 'phosphor' contacts. Luminous emissions from colour televisions are seen by the human eye in an additive process, in a manner similar to that observed through impressionist paintings which are based on the juxtaposition of coloured points (pointillism) [Til 00]. Each luminous element, called a pixel (from the contraction of 'picture element'), is in fact the product of three 'sub-pixels' (contacts placed as shown in Figure IX-I) each generating one of the three primary colours (blue, green and red). Each contact is composed differently to give the different colours: activated silver with zinc sulfide (Ag: ZnS) for blue, activated copper and zinc and cadmium sulfide (Cu:(Zn,Cd)S) for green, and activated europium and yttrium oxysulfide (Eu3+ : YzOzS ) for red.

III Electroluminescent inorganic diodes Electroluminescent inorganic diode s underwent expan sion in use towards the end of the I960s due to their lost cost (per industrially produced unit), their long lifetimes, their ability to emit light over a wide spectral range and their ease of control by a simple electronic circuit. They are widely used in domains requiring small emitters (digital displays for measuring devices and some calculators) and as calibration standards. While their individual price is low, the large number required to make an high definition screen from them makes such an overall cost prohibitive, adding incentive to finding alternative display technologies.

1 How they work In relatively simple terms, an LED is made from a directly polarised pn junction. Under the effect of this polarisation (V), charge carriers are injected from one region , where they are in the majority, to the other, where they are a minority. So electrons from the n side are injected into the p side, while holes are injected from the p side into the n side. Figure IX-2 shows this crosso ver. During the diffusion of the charge carriers, which for electrons is of length Ln and for holes is of length Lp, the minority carriers can recombine with majority carriers (minority electrons with majority holes in zone p and minority holes with majority

IX Organicstructures and materials in optoelectronic emitters 237 Zonep

Zonen

••••••••

Figure IX·2. Radiative recombination zones in a light emitting pn junction working under direct polarisation.

electrons in zone n). Usually we neglect the number of recombinations in the zone ZCE (Figure lX-2) which tends to be particularly narrow under typical tensions (V) [mat 96] . If the semi-conductor used is at direct gap (that is with the minimum conduction band and maximum valence band, in reciprocal space , at the same wavevector value), then electron-hole recombinations are facile as a third assisting particle is not required (such as a phonon for which the wavevector assures the conservation of the quantity of movement for indirect transitions-which are oblique and characteristic of semiconductors with an indirect gap such as silicon). The emission of photons corresponding to this recombination occurs with energy of the order of the size of the band gap. For GaAs (direct gap) this is equal to 1.43 eV, an emission with a wavelength ('A.) of 0.861J,m and in the near infrared.

2 Display applications To use semiconductors in displays requires the use of GaP which exhibits an indirect band gap of around 2.3 eY. Due to this indirect gap (and to reinforce radiative recombinations) impurities such as nitrogen are introduced into the semiconductor to form donor type states close to the conduction band [bre 99] . Excitonic radiative emissions can thus be generated in the green region of the visible spectrum with GaP(N) . Other colours can also be obtained, for example by doping GaP with zinc or oxygen, or by using mixtures of semiconductors such as GaAsl-xPx . In the latter case, the direct gap in GaAs increases with the concentration of phosphorus. With a direct transition, a red emission can be obtained using 40 % phosphorus, while orange and yellow emissions occur on adding nitrogen to varying compositions of GaAsl -xPx. Finally, blue emissions can be obtained using GaN or with the alloy InGaN .

3 Characteristic parameters (see also Chapters VIII and X for further definitions) The internal quantum yield, which is defined as the ratio of photons produced to the number of injected electrons, can reach 50 % in direct gap semiconductors. However, the external energetic yield (ll) which represents the ratio of luminous energy to the

238

Optoelectronics of moleculesand polymers

10 cosS

Figure IX-3. Lambert's emission represented. electrical energy supplied is typically of the order of I to 10 %. The mechanisms which result in this difference can be classed into three types : •

•

•

the absorption of photons by the material which they traverse (for which multi layer structures of materials with gaps above the emissions energy have been designed, so as to limit such absorptions); Fresnel losses which result from a difference in the refraction indices of the diode and air giving rise to the reflection of a part of the wave and its associated power; and losses due to total internal reflection when photons emitted strike the diode/air interface at an angle greater than the limiting angle (indice of diode is greater than indice of air) and remain trapped within the diode.

e,

As detailed in Chapter VIII , the luminous intensity varies according to Lambert's angular law (Figure IX-3), that is I(e) = 10 cos e in which 10 is the intensity in the direction normal to the surface and is the emission angle (again st the normal) . Therefore, if increases, cos decreases along with the luminous inten sity emitted in the same direction.

e

e

e

4 In practical terms Electrons exhibit a higher mobility than holes , and the level of electrons injected into zone p is greater than the number of holes in zone n. The result is that zone p is the principal centre for radiative emis sion s and is thus used as the emitting face of the diode (Figure IX-4) .

n SUBSTRATE

Figure IX-4. Structure of a typical LED.

IX Organic structures and material s in optoelectronic emitters ,, .:

"

I

'

239

Strong incidence ", 0 os 0,) \

! I I

! Weak incidence (0« 0 ,)

Figure IX-S. The role of the posit ion of the dome .

Finally, practically speaking, LEDs are encapsulated with a transparent resin dome which serves three functions : • • •

protection of the diode and its connectors from the exterior; reduction in the difference in indices of the diode and air, thus increasing the overall yield of the diode (reduction in Fresnel losses); and reduction of total internal reflection effects by generating an incidence angle at the dome lower than the limiting angle 81 (see Figure IX-5).

It is notable that for LEDs placed at the summit of the encapsulation dome (the continuous line in Figure IX-5), the largest part of emitted light hits the surface at an angle below the limiting angle. A diode of this type generates a greater angle of view (when looking at the diode) than a diode which has the emitting centre towards the rear of the dome, which emits light that falls on the sides of the dome with an angle close to or greater than the limiting angle.

IV Screens based on liquid crystals 1 General points Liquid crystals (LC)s were developed near the beginning of the 1960s for flat screens and, given the investment made at that time by industry (for example by Thomson LCD), this system remains the most widely used in technologies such as watches, calculators, computer screens and portable telephones. World wide, the market for LC based screens is valued at around 30 billion dollars, just below that of cathode ray tubes. The liquid crystal state is a mesophase, a state in between those of a crystalline solid and an amorphous liquid [Til 00]. There are three main types of mesophase: • • •

Timmerman's globular molecule s which give rise to plastic crystals ; discotic, resulting from flat molecules ; and calamatic, resulting from long molecule s which can be in piled layers (smectic LCs) or grouped in clusters (nematic LCs) in which the molecules are aligned parallel to one another (depending on thermal effects) .

240 Optoelectronics of molecules and polymers

2 How liquid crystal displays work Liquid crystal displays (LCD)s operate by using two basic properties of LCs: • •

the molecules can modify the direction of polarisation of an incident wave of light; and the molecules can be easily moved by the application of an external electrical field, which interacts with an orientated assembly of molecules and their orientated dipole moments (unlike thermal agitation which randomly moves individual dipoles). Even weak fields can vary by a considerable degree the orientation of the molecules. In practical terms, there are two main types of system: [Car 84]

a 'Guest-host' type displays These displays use dichroic dyes (denoted 'guests' ) orientated by the crystal liquid molecules (the 'hosts') . The elongated dye molecules absorb the component light vector parallel to the molecules' axes (see Figure IX-6) but not the perpendicular components (due to dichroism) . Depending on the direction of the molecules , determined by an applied electric field, an incident wave polarised parallel to the molecules (which have an initially homogeneous arrangement) would be reduced or become non-existent. It is in this manner that displays can be prepared using nematic LCs with dichroic dyes introduced into sandwiched cells, constituted of glass plates, 5 to 200 urn apart, covered with the transparent conductor ITO (a mixture of indium and tin oxides). In reality, two optoelectronic effects may be observed: the first corresponds to an inversion of contrast (from an initial homogeneous orientation as in Figure IX-7-a) and the second a normal contrast (from an initial homotropic orientation as shown in Figure IX-7-b).

b Displays made with helical nematic liquid crystals These are the most widely used and are based on the rotation of the plane of polarisation of light by 90 0 by 'twisted' nematic liquid crystals, in a configuration shown in Figure IX-8.

wave Figure IX-6. Usinga dye to modify wave polarisation.

IX Organic structures and materials in optoelectro nic emitters

Incident light

lncideut Iiaht

( )

Polarising filter a ;5 GIll! cover

•

I

Glass cove r

- : .- : .

-=-

Dye

illP l ~

Polari sing filter axis

I

~

Layer of ITO Liquid CI"),tltl/

24 1

_

1..a\'erofITO

Uq";d'~ ~I./" D )' c

r = = = = = =1' ••

I

I

_

I

.....

Light cut off

I •

Light tran:millcd

Incident light

Incident light

(b)

Polari sing filter axis Glu ' cover

I

I yer of ITO

Liquid Cf)"tHI/

Dye

Polartslng fllter axis GIll! cover I

•

I I I

I

~

~

I

.

I

. ..

••

• I

~

Layer of ITO Liquid cr)"lal/

_

- : .- : .

L-

~I

~

Dye

•

Light cut off

I rausruitted light Figure IX-7. (a) Initially homogeneous orientation: inverted con trast; and (b) Initially homotropic orientation: normal contrast.

The liquid crystal is encapsulated between two glass sheets around 10 u.m apart each covered by a transparent ITO electrode, The glass sheets are surface treated so that on contact with the liquid crystal, the molecule s of the latter align themselves along a given direction . In addit ion, the directions of molecules at the two electrodes are such that there is 90° between them. As shown in Figure IX-8-a, a polarising filter, on the upper side, allows only incident light orientated parallel to the LC molecules through. Throughout the thickness of the LC layer, the molecules turn this single light component so that on reaching the lower face, the light is polarised parallel to the seco nd polarising filter. Thus, in this relaxed state, the system appears transparent. Once a tension is applied, using the electrodes detailed in Figure IX-8-b, the electric field orientates the LC molecules so that they are parallel to the propagation component of the incident light. The LC molecules no longer turn the direction of the polarised traversing light and therefore, when the light reaches the lower face, it arrives with a component perpendicular to the second filter and the system now appears opaque.

242

Optoelectronics of molecules and polymers

Polarising filler

layer used to align LC

molecules

LC molecules

layer used 10 align molecu les

L~~~""I lII!l

Polarising filter Light transmiued: 'clear" . n...,n

~

Opaque system: black screen

Figure IX-8. (a) Without polarisat ion oflight in between electrodes , LC are in helical form and system allows light to pass : observer sees transparent system; (b) With electric field applied (~l V), the initial orientation of molecules is disturbed and light is no longer turned to pass through the second polarising filter: the observer sees an opaque system .

The response time of such a system can be of the order of several tens of milliseconds, a value which permits the use of these materials for display technologies (which rely upon the persistence of perception of light of the human eye).

3 LCD screen structure and the role of polymers Screens are made up of a matrix of rows and columns of pixels (to give a matrix screen) . Colours are obtained by dividing each pixel into three sub-pixels (red, green and blue), each of which can be individually addressed. A side view is given in Figure IX-9.

DilTuser Polarising filter 3 comma nding electrodes Commanding column_~t;J."'-=~I!IIt:;lI!l!l. Comma nding row Liquid crystal Common electrode Layer of coloured filten>--fL-Polarising filter Glass substrate Figure IX-9. Cross section of a LCD pixel.

IX Organic structures and materials in optoelectronic emitters

243

Polymers are involved at several levels in the manufacture of LCD screens and help in reducing their energy consumption and facilitating their use, by introducing qualities such as low weights, flexibility and so on [bro 01]. They are used in particular in the following ways : • • •

•

photosensitive polymeric resins used in the photolithographic processes; in cons tructing the screen and the walls of each cellu lar pixel which contains active materials ; in aligning and orientating layers ofliquid crys tals, with the recent use of polymers that have improved the viewing angle and have allowed , using a stamped film layer of polyimide, control over the alignment of molec ules in each sub-pixel; and optical functionalisation of screens, whereby stretched polyvinyl based polarising filters are protected by cellu lose triacetate, which are filters coloured by photosensitive acrylates. Diffusion of light can be obtained using a stretched polyester film. Indeed, mixtures of birefringent polymers can be used to generate reflections by diffusion of otherwise untransmitted, polarised light permitting a recovery of this unused light. Using these techniques has allowed an increase in the luminance of screens, from around 200 cd m- 2 for the more classical systems based on backlighting to around 500 cd m- 2 , a value similar to that obtained with cathode ray tubes.

4 Addressi ng in LCD displays [Dep 93] Multiplex addressing (reaching any single pixel at a given moment) necessitates a system for addressing each pixel in a display[bar 00] . In passive matrixes , as shown in Figure IX-1O-a,each pixel is addressed via the use of a system of electrodes placed in rows, placed for example on the upper glass plate, and in columns, placed for

(a)

Impulsc tensions applied at columns

u unnuuu .-

R~

impulse tcjlot

-

-

-

-

-

.-

I I I I I I Selected ells

Fig ure IX-l O. (a) Passive matrix addressing; and (b) Active matrix addressing.

244 Optoelectronics of molecules and polymers example on the lower glass plate. At each intersection is a pixel. The surface of the screen is swept in a 60th of second by the electrical excitation of each row while each column is subject to a tension sufficient to produce the required excitation of each pixel. Pixels which are not selected for illumination are subject only to a low or residual tension, the latter of which increases as the resolution of the screen is increased . Indeed, to increase the resolution, along with the number of rows, the tension needs to be increased to maintain an overall average luminance of the order of 200 or 300 cd m- 2 ; any increase in resolution also necessitates a decrease in contrast as otherwise some pixels would be illuminated simply by going over the threshold tension . In order to resolve the problem of decreasing contrast with increasing resolution, active matrixes are used which have individual command transistors associated with each pixel. These transistors are called thin film transistors (TFf) and consist of two electrodes for the entrance and exit of current which flows under the control of a third gate electrode . Hydrogenated amorphous or polycrystalline silicon has been widely used although electroactive polymers have been used as organic electronic / optoelectronic materials . Examples of polymers developed for such use are poly(para-phenylene vinylene) (PPV) [gre 95] and polythiophene, and such materials being more and more widely used. A more detailed description of the function of organic transistors can be found elsewhere [hor 00].

5 Conclusion In conclusion, the technology used for LCDs is now very wide spread and has given rise to screens which operate at low power rates. One of the major limits of the technology, however, is the viewable angle of LCD displays , at around 30 0 to 40 0 , which results from the technology itself (molecular alignment forming a directional guide), although some progress has been made in improving this [jon 99]. Hitachi and NEC have realised displays with excellent viewing angles by using coplanar electrodes which control, in the same plane, the optical commutation of the liquid crystals. In addition, very wide screen displays, for example for use as wall televisions, have been prepared (from 1990 on by Tektrononix) by replacing the TFTs by plasma canals which successively trigger each row of pixels.

V Plasma screens Plasma screens were developed around the beginning of the 1970s following the demand in flat screens. Monochromic plasma screens operate by emitting orangeyred light from ionised gas, much in the same way as neon tubes[dep 93]. The gas, confined within a series of small cavities each positioned at the intersection of upper and lower electrodes (which determined the size of the pixels, as shown in Figure IX-II), is excited to the plasma state by applying a certain, selective tension. Displays made using this technology exhibit high brilliance and good lifetimes and stabilities. However, they require a considerable amount of energy to operate (and run

IX Organicstructuresand materialsin optoelectronic emitters

245

Insulator-i~:;::=~~~ Upperelectrode-s Magnesium oxide Pixel or sub-pixc eparator Neon + xenon Phosphor -t:::::~~-==t=l Lowerelectrode Glas

Figure IX-H. Cross section of a plasma screen.

at around 200V). Other colours can be made by indirect excitation of luminescent substances (again using primary colours) with a plasma that emits in the ultra-violet, an example of which is a gaseous mixture of neon and xenon . The geometry of each sub-pixel, which emits either red, green or blue , is shown in Figure IX-II. The cavity, containing the excitable gas is covered with one of three phosphors. The layer of magnesium oxide is there as a dielectric to reinforce the electric field within the cavity [Til 00].

VI Micro-point screens (field emission displays (FED)) FEDs are microelectronic components which approximate to miniature cathode ray tubes [Bre 99]. In a FED, electrons are emitted from a cathode and collide with a phosphor to give an excitation and thus a luminous emission. Cathodes are assembled above an insulating layer, as shown in Figure IX-12 . Each pixel contains three phosphors and three metallic electrodes which play the role of gates for each colour.

anode

gate electrode

gate electrode

Figure IX-12. Cross sectional structure of a micro-point screen (FED).

246 Optoelectronics of molecules and polymers A pixel is activated firstly by the application of a threshold tension (cathodic tension). Any current that flows is controlled with the help of a variable tension applied via the gate electrode. This controlled current of electrons is then accelerated towards the anode . The primary electronic current, which has a more or less radial propagation, is then focussed by use of gate electrodes placed about the micro-point cathode so that the electrons reach the anode in a nearly parallel beam. FED based screens exhibit several advantages over cathode ray tubes and vacuum tubes: • •

they can produce a higher current density during thermo-ionic emission than that encountered in vacuum tubes ; and they do not require filament heating as electrons are generated by a field effect obtained at the extremity of a micro-point cathode, where the field is sufficiently intense to develop electrons through tunnelling effects across a reduced, narrow potential barrier between micro-point tip and the vacuum (see also Annex A-7, Section III).

With respect to LEDs based on semi-conductors, FEDs also present some advantages: • • •

no losses due to dissipation ( i.e. no colli sions between electrons in the vacuum while electrons may collide with a semi-conductor lattice); operation practically independent of temperature; and no perturbations due to ionising rays.

However, FED based screen s do pose some disadvantages, for example the minimum distance required between anode and cathode to reduced sparking, and the non-directed emission of electrons tending to limit the obtainable degree of resolution.

VII Electroluminescent screens 1 General mechanism Electroluminescent screens function using Destriau's effect [mat 96], in which radiative recombinations are generated by the application of an electric field to a powdered semiconductor dispersed within a dielectric matrix, as shown in Figure IX-13-a . The semiconductor is normally the mixture of an inorganic material from group s II-VI, with a relatively large forbidden band of around 2 to 3 eV, and an activator made of a transition metal (typically copper of manganese) or a rare earth . In addition, fillers such as NaCI or KCI may be used as they are easily melted and on cooling they favour the formation of crystals and the insertion of the activators . The electric field plays the role of carrying the concentration of free charge carriers above its equilibrium value; a return to equilibrium results in recombinations and, in the case of radiative recombinations, the emission of photons. The concentrations, outside of equilibrium, can be generated (and can accumulate) by applying a strong enough field. In general :

IX Organic structures and materials in optoelectronic emitters

(b)

;1

Sn02

Dielectric

/V

ZnS

u

(a)

ZnS -

247

·C /

....o

~

o

Q

lctallic electrode

BV

E applied

Figure IX-13. (a) Structure assisting the Destriau effect; (b) Mechanism of charge injection into electroluminescent (ZnS[Cu]).

•

•

when E, < 108Ym- l , shock ionisation proce sses (with Wionisation = q E, I, where I is the distance covered by an accelerated carrier between two impacts) shift electrons into the conduction band of the semiconductor from valence or impurity bands ; or when E, > 108 Y m -I , carriers can be injected through tunnelling effects at the interfaces (see Figure IX-I3-b) to increase the density of carriers available through shock ionisation.

The component structure of electroluminescence will vary depending on the type of applied tension (altern ating or direct).

2 Available transitions in an inorganic phosphor In the band scheme of an inorganic phosphor there are permitted bands associated with activators (otherwise called luminescent centres when they induce radiative transitions). There are two possible types of centre within a phosphor: •

•

those which introduce into the forbidden band localised levels close to the valence band ; this is the 'classic' model of Cu centres in ZnS which give rise to a level at several tens of eV above the valence band that has a high probability of capturing holes ; and those which reinforce both the fundamental level I close to the valence band and the excited level II just below the conduction band. Levels I and level II exhibit high probabilities of capturing holes and electrons, respectively. An example of such an activator is CuCI, [dep 93] although Mn3+ is more widely used along with rare earth ions [mom 95] .

There are other bands, very clo se to the conduction band , which also exist. They are associated with structural defaults (and thus the preparation of the phosphor) and act as trap levels . Electrons caught with in these bands may be thermally released with a probability given by the Randall-Wilkins law, which supposes that electrons within the traps have a Maxwell type distribution of thermal energies (see also Chapter VI,

248

Optoelectronics of molecules and polymers

Section IV-4). It can be written in the form p

= Vo exp( -Us/kT),

where Us is the energetic depth of the trap with respect to the edge of the conduction band and vo,a priori, is a constant which takes on a value of the order of the frequency ofphonons (;::::;10 12 sec-I according to Mott and Gurney) although it does tend to vary with temperature and typically has been found to have a value around 108 sec"! in phosphors [ran 45]. Figure IX-14 shows a representative energy diagram for activated phosphors [fri 85]. There are various transitions which can be envisaged: • • • •

•

A: an activator is excited, recombination occurs and an electron returns to the fundamental level; B: excitation, in the conduction band, of an electron at an activator and a recombination with another activator centre; C: same as B but with a delayed recombination due to the passage of electrons via trap levels, which they escape through thermal energy; D: band to band generation due to energetic beams (>2 to 3 eV) whereby a free hole generated from the valence band is displaced until it recombines with an electron in the neighbourhood of an activator. The latter acts as a radiative centre, and an activator electron falling from the valence band to recombine with a hole results in a positive ion at the activator itself, which may in tum may capture, radiatively, an electron from the conduction band; and E: the same process as D but a similar delay to that in process C in which electrons are delayed by traps .

Generally speaking, the term 'fluorescence' is used for immediate relaxations (for example, cutting out trap levels and within a time 1: ;::::; 10- 8 s) while the term 'phosphorescence' is used for recombinations which occur after a delay 1: > 10-7 s. Comment It is worth noting that while an increase in temperature may help detrapping of electrons , it can also lead to an extinction of luminescence. This is due to electrons excited from the valence band to empty levels of the activation centres,

Conduction band Traps Activators' excited levels

u

E~:::':-='::~:::'

.:- ::::::.~::: '~:::-:'': ::~ ..::.:~~:: ::::: :: ::~:::.:: BCD

Act ivators' fundamental level

E

_ ._. ._. _. Valence ham!

....._---------~----

Figure IX-14. Energy levels in a semiconductor (II-VI) activated by transition metals or rare earths.

IX Organic structures and materials in optoelectronic emitters

249

Table IX .t A. (nm)

ZnS (blende) ZnS (wurtzite) CdS (wurtzite) ZnSe (blende) CdSe (wurtzite) ZnTe (blende) CdTe (wurtzite)

341 335 510 477 711.5 578 871

E (eV) 3.64 3.70 2.43 2.60 1.74 2.15 1.42

which can therefore no longer act as recombination centres, or excitation of carriers simply to be trapped non-radiatively at extinction centres.

3 Characteristics of inorganic phosphors from groups II-VI [cur 60] Groups II-VI have a forbidden band of energy EG which is close to that of photons in the visible range . The materials most widely used are ZnS (which as zinc blende in Europe or sphalerite in the USA is of a cubic structure, or as wurtzite is of a hexagonal structure) and CdS. Table IX.I below gives the values of EG and the wavelength of corresponding photons for the principal mixtures used. The emission band to band transitions observed for these phosphors actually appear unrealistic, however, it is the introduction ofluminescent centres , as mentioned above typically based on copper or manganese, which permit a circumnavigation of the problem. In Figure IX-IS there is shown the relative position of energy levels associated with copper luminescent centres introduced either into ZnS or CdS . In both cases there are two principal emission bands. In the case of ZnS(Cu) there is one in the green at 527 nm and one in the blue at 445 nm. In classical terms, photoluminescence occurs when centre excitation (by UV light) results in the generation of valence band holes, into which there is then a fall of electrons from luminescent centres (see Figure IX-16-a).

(a)

..

ZnS CB

I

3.70 cV 12.37 cV 2.79 cV I 15Z.\ nm ! (.usom! Cui _t'---_ _ Cull ......:..._-

VB

3cV

VB Figure IX-IS. Energy levels in: (a) ZnS(Cu) ; and (b) CdS(Cu) .

250

Optoelectronics of molecules and polymers

(a) I, citation

t ','IlUl:

electron . CB

• o

o

VB ..,.- - - -Hole

(b) Radiative emission .1uo!J'S'

Pho
•

•

CB

[T o

_....:. 0_ _

VB

Figure IX-16. (a) centre excitation; and (b) radiative emission,

Radiative emission though corresponds to the recombination of conduction band electrons (from group II-VI elements) with holes localised at luminous centres (through process D indicated in Figure IX-14) . If the transition occurs directly, without passing via traps situated just below the valence band, then the emission occurs as fluorescence, however, if electrons relax on passing through trap levels, then the emission occurs as phosphorescence, as shown in Figure IX-16-b. Phosphors based on ZnS are not easy dope so that they are of n or p type, [page 79 of mom 95] and this sort of material cannot be used in a pn type LED.

4 Electroluminescent think film displays: how they work with alternating current A general abbreviation often used in this area is ACTFEL standing for 'A.C. supplies in a Thin Flim Electroluminescent displays' [bre 99]. Typically, thin film electroluminescent displays are based on an insulatorsemiconductor-insulator structure, as detailed in Figure IX-17-a .

D~elecltl c

layer Tunn elling effect injects electrons into CB

------I

ZnS(Cu)

I

(a)

Applied electric field

Figure IX-!7. (a) The different layers in an ACTFEL screen; (b) Band positioning on application of an intense electric field.

IX Organic structures and materials in optoelectronic emitters

251

The active ' phosphor ' layer (ZnS(Cu)), which is about 700 nm thick, is set between two transparent dielectric layers, them selves each about 400 nm thick.The latter layers are in cont act with electrodes , one aluminium, the other transparen t ITO deposited on a glass substrate. Thi s structure resembles that of capacitors in series, and permits the generation of a relatively high tension at the active ZnS(Cu) layer from a low tension at the electrodes by using the low capacities of the thin dielectric layers. On application of a strong electric field (E a ) the high degree of band 'b endin g' of the active layer allows electrons, which would otherwise be trapp ed at interface states, to pass throu gh to the conduction band by tunnellin g effects (Figure IX-17-b). Electron s cross the active layer by collecting energy along their pathway of distance I. With W = e E, I, and starting at the threshold energy, these electrons can transfer by collision a high enough amount of energy to excite the luminescent centres with a return to equilibrium throu gh lumin ous radiation. Given that the concentration of centres is rather low, of the order of lO IS cm- 3 , and that of the phosphor hosts is of the order of 1023 em - 3, the actual available surface for generating impact excitations is very low. Accordingly, only a small proportion of the injected electrons result in the excitation of a centre while most reach the opposing electrode to give rise to a space charge which can affec t following charge injections into the comp onent. The application of an alternating current, which continually reverses the sense of the applied tension, can use these electrons to aga in excite the lumin ous centres in the reverse direction (these electrons accumulate with electrons newly injected throu gh tunnel effec ts from levels localised around the seco nd electrode). The overall brilli ance of these screens remain s relatively low, principally because the source of the elect rons, the semico nductor-insulator interface , has at any single moment a relatively low concentratio n of electro ns and leading to a low availability of electrons for impacts with lumin ous centres. However, the screens are relat ively robust and are widely used as monochrome displays in military applications. In addition, different colours can be prepared. On using manganese ions (Mn2+), introduced by way with zinc sulfide (Mn 2+ : ZnS), a yellow emission at 585 nm can be obtained, a result of the d-d transitions with the breakdown of d-orbitals by crystal potentials. Other colours can be generated from f-f orbital transitions (due to incomplete internal, atomic 4f layer s) in rare earth s. For example, red can be obtained with europium [Eu2+: CaS], green with terb ium [Tb 2+ : ZnS] and blue with thulium [Tm3+, F- ZnSl (see also Chapter VII, Section II for more details).

5 Electroluminescent devices operating under direct current conditions Here, the structures used are either of the metal-in sulator-sem icondu ctor (MIS) or the Schottky barrier type. In both syste ms, the electron source is a metall ic electrode. The structure shown in Figure IX- 18 operates by the passage of electrons throu gh the insulator barrier from the semico nductor valence band (VB)- typically CdStowards a positively polarised metal electrode. The mechanism is equivalent to the

252 Optoelectronics of molecules and polymers

Figure IX-IS. MIS structure with CdS semiconductor.

injection of holes into the semiconductor VB with the following recombination of VB holes with conduction band (CB) electrons. This structure does, however, present two inconveniences: •

•

in the recombination zone, which is placed close to the interface , the interference of defaults, which act as trap levels, results in non-radiative recombinations; and the efficiency of hole injection is very poor due to the competing flow of electrons through tunnel effects from the semiconductor CB to the metal.

In the case of the Schottky junction, an inverted tension is applied (i.e. the metal is negative if the semiconductor transports electrons). Once the tension is strong enough, electrons may be injected through tunnelling effects (barrier triangulation) from the metal to the semiconductor. Thus , a considerable number of electrons can be injected-more than is possible using an alternating tension-and can as carriers collide with luminescent centres such as Mn (transition metal) or rare earths . These devices can function in both continuous and pulse modes, although in the former mode their lifetimes are shortened by a increase in resistivity, a problem which is reduced by using the pulsed mode. (An alternating mode does not give the same problems as it relies upon the structural capacitance). The quantum yields of these devices in the continuous mode is rather poor (n ~ 0.1 %) because of: • • •

the difficulties in obtaining a high enough number of hot carriers to excite the luminous centres; a low, effective sectional area of centres for collisions; and a limited concentration of luminous centres (from 0.1 to 10 %). Above a certain concentration, however, extinction of luminescence occurs due to the crossed relaxation by transfer of energy between luminescent ions (Chapter VII gives further details).

The main advantage of these systems , over those operating under alternating current, is their low consumption of electricity and their relatively easy construction due to their simple structure.

IX Organic structures and materials in optoelectronic emitters

253

VIII Organic (OLED) and polymer (PLED) electroluminescent diodes 1 Brief history and resume The ability of organic materials , both molecular and macromolecular, to be deposited on virtually any substrate, even flexible ones, has solicited the long running and ongoing research into their use as replacements for inorganic semiconductors in a wide range of applications. For a considerable time, the necessary tension required to produce electroluminescence (the required work voltage) of these organic materials was found to be too high for their general use. An example is that of anthracene which required, because of the thickne ss of the crystal s used, an application of around 100 V to operate [pop 82]. However, in the last 10 years, the emergence of new materials based on molecules such as 8-tris-hydroxyquinoline aluminium (Alqj) [tan 87] and on rr-conjugated polymers such as poly(para-phenylene vinylene) (PPV) [bur 90] has got around this problem and has allowed the formation of thin films by evaporation and spin-coating techniques, respectively . More recently, there has been tremendous growth in this technology, stimulated by the economic gains available in the display market. Companies of note are Kodak and Uniax in the States, Cambridge Display Technology (CDT) in Great Britain, Pioneer in Japan, Philips in Holland, Siemens in Germany and Thales and Thom son Multimedia with the CEA and Leti in France.

2 The two main developmental routes There do remain, however, certain problems to overcome before these materials can provide the qualities required for full industrial exploitation. Most important is the fact that, until now, most organic materials have displayed relatively short lifetimes, varying from several hours to several hundred hours , although in some cases several thousand hours . This limitation results just as much from the deterioration of the material s themselves as the degradation in the various interfaces and the electrodes, and depends heavily upon the configuration used in each device [sly 96]. In addition, the material of choice for the emitter has yet to be determined [bar 00], and could be either based on molecules (for example Alq, or more generally speaking q3 type complexes with group lIla metals such as AI, Ga or In) or based on polymers (such as PPV and its derivatives, poly(para-phenylene) (PPP), polythiophene (PTh) or poly(vinyl carbazole), to name but a few). Each material presents both advantages and disadvantages. Oligomers, polymers made up of just a few monome r units, present an interesting advantage in that the maximum emission wavelength can be altered with chain length, given that with changes in their length, their optical band gap also varies [had 00]. Polym ers, meanwhile, generally present a better thermal stability and can be easily processed to give large surface area films, although small molecules can be spread onto large, flexible supports using the 'roll on roll' technique [bur 97]. Materials based on molecules are generally more organised, charge mobilities are higher and function at lower operating tension s. In addition , they are readily purified, resulting in lower numbers of reactions and diffusions at the electrodes.

254

Optoelectronics of moleculesand polymers

For the present moment, both technologies based on molecules or on macromolecules remain important in the fabrication of organic electroluminescent diodes. The terminology used for devices depends on the materials used: the term Organic Light Emitting Diodes (OLEDs) is widely used for devices based on small molecules and Polymer Light Emitting Diodes (PLEDs) is used for those based on macromolecules. For the latter, another term although less used is Light Emitting Polymers (LEPs).

3 How OLEDs function and their interest As detailed above, OLEDs have the advantages of low power consumption (requiring operating voltages of only 2 V) and ease of fabrication (easily formed into thin films or layers). In addition to which, relatively high quantum yield s have already been obtained. Their theoretical maximum yield is 25 %, however, this figure may be eclipsed by recent methodologies which try to circumnavigate selection rules. Given these qualities, it is not surprising that we are now witnessing an exponential growth in the technology and its market [ole 01] . In the following Chapter X, we shall look in more detail at how these organic LEDs actually work, both in terms of theory and practise. At the most fundamental level though, the origin of the optoelectronic properties of these organic solids has been given in Chapters VI and VII. In very general terms, the study of photometric characteristics of OLEDs, in particular their quantum yields but also their lifetimes, are guided by an understanding of the underlining mechanisms of charge behaviours and recombinations (radiative or not) within the organi c material. In classical terms, an OLED can be schematised as shown in Figure IX-19 . The structure used for an organic diode is very different from that used for an inorganic diode. It is not possible to use organic semiconductors in their doped form , which would permit the fabrication of pn junctions similar to inorganic diodes, as doping agents tend to act as extinction centres, 'killing' radiative luminescence [ham 96 and hay 97]. In organic materials, what doe s happen is that the low mobility of the charge carriers is exploited, so that when two charges are injected into the emis sive

Electron injection layer Emission

layer( Iq3. PPV )

Hole injc tum layer ITO (Anode)

/-- - - - --:=::::\-- - - -

Emitted light

Figure IX·19. Typical structureof an electroluminescent organicdiode.

IX Organic structures and materials in optoelectronic emitters

255

layer and come into close proximity, their recombination is inevitable. This is in contrast to inorganic systems in which charge carriers have such a high mobility that recombination centres are required to realise a satisfactory level of recombinations. It is due to the very low mobility of charge carriers in organic materials that there is this gain in a high concentration of optically active recombinations. It is thus useful to study the how organic diodes function in terms offour successive steps : • • • •

injection of electrons and holes at the cathode and the anode , respectively; their transport through one or more organic layers; their association to form a exciton quasi-particle; and the relaxation of the quasi-particle resulting in luminescence.

As detailed in Chapter X, the models so far defined for each of these steps remain controversial. What is probably true is that each type of material , and each type of structure, follows a different system model. For example, current-voltage characteristics are often studied , and explained, in terms of injection either through Schottky effects or through tunnel effects. They can also be interpreted through the so called 'space charge limited' (SCL) current model. In order to reduce the threshold tension for a particular diode, and depending on the dominant process, we should look more to varying the metal of the electrode and the mobility of charge carriers within the organic layers. In Chapter X, we shall concentrate on this physical aspect of the functioning of electroluminescent diodes . The chemical and physico-chemical aspects of these devices have already been covered in depth elsewhere [bar OOa, den 00 and kra 98] so will not be covered in any great depth here. We shall finally concentrate on device applications currently being developed, with special regards to the area of display technology.

x Electroluminescent organic diodes

I Introduction To have an idea of the vast number of articles publish ed regularly on the subject of electroluminescent organic diode s, it is worth looking at the journal ' Synthetic Metals ' , in particular issues devoted to the ' ICSM' conference. While a large number of papers are published in this journal, it shows only a small fraction of all the material available . This Chapter can therefore only limit itself to a very cond ensed view of the subject. Following on from Chapter 9 where the princ iple upon which these device s function was presented, this Chapter : • • •

compares empirically obtained results with those expected from theories detailed in Chapters VI and VII; considers the yields obtained with organic light emitting diodes (OLED) s and the strategies used to improve these yield s; and details the actual and possible future applications of these device s.

This Chapter will not go into discussions on the origins of electroluminescence in organic materials (radiative transition s) as Chapter VII (Section V-4) has already covered an interpretation of the obtained spectra, while Section VI of the same Chapter detailed the nature of excitations (excitons). We have also seen how electron-lattice interactions playa role in these properties, without forgetting the limits of the model, originally proposed for small molecules, when applied to polymer s. For now, we shall ju st give the simple example, as under stood by Burrows et at. [bur 96], that electroluminescence in molecular organic light emitting diodes (OLEDs) is due to the generation and recombination of Frenkel excitons within the Alq3 layer. This was shown by observing lumine scence spectra of Alq3 in solution (in dimethylformamide), which , unchanged from those of solid, vacuum evaporated Alq3, indicated that the recombinations were independent of the molecular environment and originated from excitations localised at individual molecules. These excitons come from both electrons and minority hole s, which control luminescence inten sity, and diffuse through layers which will accept their injection. We shall look at these

258 Optoelectronics of molecules and polymers layers in more detail later on in this Chapter. The electron s are localised at traps, which are themselve s distributed within a gap of average depth E, < ELUMO' These levels come from electron s, which act as polaron s initially placed within the LUMO conduction band (CB) and localised (trapped) at molecule s with relaxation s generating a level with depth Et • Indeed , the shift in the spectrum of electroluminescence with respect to that of absorption can be interpreted as a Franck-Condon displacement, due to the change in the conformation of excited molecule s, while the width of the electroluminescence spectrum can be attributed to strong exciton-phonon coupling.

II Comparing electronic injection and transport models with experimental results 1 General points: properties and methods applied to their study a Properties From the first studies performed on organic solids (using anthracene), it was observed that various regime s of behaviour could exist, especially for example with respect to applied tension. Chapter VI gives further details . However, with OLEDs , clean transition s between different regimes are generally rather rare. Experimentally, we tend more to see only slight variations from the slopes predicted by laws detailed in Chapter VI. Two example s are used in this Section: for the small molecule s, a derivative of Alq3 ; and for the polymers, poly(para-phenylene vinylene) (PPV). The structure in which these materials are presented is simplified to anode/organic solid/cathode, in which either Alq3 or PPV is the organic solid acting as a layer into which both holes and electron s are injected and electroluminescence occurs. There have been published propo sals for distinct mechanisms for each of the material s under study here, respectively, [bur 96] and [par 94], with following and differing proposals, again respectively, [ioa 98] and [blo 97a, blo 97b and blo 98]. More complicated multi-layer structures are discussed in Section V.

b The nature of the electrical measurements and some practical precautions Given that these materials are relatively resistant, it is worth bearing in mind that any measurements must take account of their relatively long relaxation times, of the order of a millisecond to a second, rather than of nanoseconds or less [kar 97]. In practical terms, we can observe in the first characteri stic I(V) curve that there are peaks in the tension which may appear randomly , only to be unobservable on the second trace. These peaks may be due to the filling, during the first trace, of certain deep traps which, statistically, do not empty themselves-and therefore do not need refilling-so that they effectively disappear from the second I(V) trace. It is generally the second curve which is shown.

X Electroluminescent organicdiodes 259 Classically, characterisations are performed by measuring the current flow across a sample to which is applied an increasing tension . The intensity of any electroluminescence is simultaneously recorded (initially with respect to luminance) and the following identification of charge transport mechanisms is made by comparing the empirically obtained I(V) curve against those predicted by theory . The results , interpretations and discussions issuing from this process are presented in the following paragraphs. It is worth noting, however, that there are other important electrical characterisations which may be performed: •

•

•

time of flight measurements which indicate charge carrier mobility (for example, with PPV, at an applied field of 105 V cm- 1, a mobility 11 ~ 10- 5 cm 2 V-I s-I has been observed); measurements of thermally stimulated detrapping, using thermally stimulated currents (TSC) or thermoluminescence (TL), which permit studies of the population of traps and their origin or eventual depth . Using these methods , the depth (U) of traps in Alq3 were determined as U ~ 0.2 eV [for 98] and in PPV as U = 0.35 eV for traps in the volume and U = 0.68 eV and 0.82 eV for the levels of traps at the interface, whose values depended on the polarisation tension [ngu 01]; and dielectric studies which permit measurements of metal-semiconductor barriers for Schottky junctions (using the curve I/C 2 = f'(Vj), or the evolution ofrelaxation phenomena with temperature and frequency to resolve both real (E') and imaginary (E") dielectric permittivities.

Different layers of dielectrics and their interfaces can be represented by equivalent circuits of resistors and capacitors, the values and connections (series or parallel) of which are varied until a near approximation is obtained [jon 83]. For the phenomenon of ideal relaxation, or a Debye dipolar relaxation, for the curve E" = f'(s') we have a semi-circle or flat half-circle (as in the Cole-Cole diagram which corresponds to a distribution of relaxation times), and the presence of a (default) interfacial layer can be characterised by an oblique Cole-Davidson type arc [ngu 01]. While relaxation phenomenon in the volume are dependent on temperature, those due to interfacial layers (next to empty space) exhibit a behaviour practically independent of temperature, as we have already seen in Alq3 . The same material has also exhibited a Debye dipolar absorption with a relaxation energy (U) around 2.3 eV,a value associated with the depth of trap levels in the volume of the material ljeo Olb] . Appendix A-IO gives further details.

2 Smallmolecules (Alq3) a A study of interface barriers in a standard structure oflTO/Alq3/Ca a Principal points The simple lTO /Alq3 /Ca structure simplifies the study of properties of charge injection and transport. The electroluminescent properties, with respect to an electroluminescent diode are, however, relatively limited as although injection, transport and radiative recombinations occur readily in Alq3, the same is not

260

Optoelectronics of molecules and polymers

true for the injection and transport of holes, which control the radiative emissions (thus the necessity of a specific layer added to the anode side in device fabrication) . Thermoelectronic or Schottky [cam 99] or Fowler-Nordheim electrode emissions are generally accepted to be responsible for injection at contacts, when the latter are not ohmic. When the height of the apparent barrier is above 0.6 eV, in general, the current within a structure is thought to be governed and thus limited by these electrodes [she 98]. Once the barrier is lower though, the current is only slightly dependent on the electrode emission (as the contact is on principle ohmic) but is heavily dependent on the volume and follows the laws we established in Chapter VI, where at low tensions the ohmic law reigns and at higher tensions the SCL and TCL laws play their roles. It is interesting to note here how there is a considerable gap between language and reality: an ohmic contact actually results in a non-ohmic conduction law of the type I ex: V'" (with m :::: 2) and is characteristic ofTCL or SCL laws, which reign over virtually all values of tension. The SCL law is exceptional in that between the values of around 0 to I V, typically, the density of injected carriers remains lower than that of thermally generated carriers (which is such that no = N, exp[ -Ec - EF/kT]) . It is only in this minor domain that ohmic characteristics are retained. We shall denote the workfunctions of ITO, calcium and Alq3 as WITO , WCa and WAlq3, respectively, and the electron affinity, the potential ionisation energy and the size of the forbidden band of Alq3 as XAlq3 , IpAlq3 and EGAlq3, respectively. As empirically derived values , we have WlTO = 4.6eV, WCa = 2.geV, XAlq3 = 3.3 eV, EGAlq3 = 2.6eV, and thus IpAlq3 = 5.geV [values for Alq3 obtained from sch 95]. On taking WAlq3 = XAlq3 + (E, - EF), a value calculated from the expression (E, - EF) = kTLn(Nc!no) using Nc = NLUMO = 1019 cm- 3 and no = 1011 cm- 3 (for Alq3 see [bur 96]) , we obtain (E , - EF) = 0.5 eV, that is WAlq3 = 3.8 eV. Figure X-I shows the electron levels prior to the application of a tension . Once contact is made with the electrodes, the positions of bands are adjusted by the alignment of Fermi levels, as shown in Figure X-2 for Vapplied = O. The values of the diffusion potentials at the two interfaces adjust so that Wdiff = WITO - WAlq3 and W~iff = WAlq3 - WCa. In general, we assume that for these organic solids the bands are rigid (oblique, thick grey lines in Figure X-2) and correspond to the metalinsulator-metal (MIM) structure in which there is a constant field across the insulator.

Vacuum reference level

--- --

Wm>=4~

I

-------

---

-- ----

3.3

\ lqJ

r.vc.

W AIqJ = 3.8 I I'AI J= \

E.·

I ITO

r.&:- ~:d.=. 0..5J.f;1 ...... Alq, ~;Alq) =

... _----

.9

=

2.9

E.·

Ca

2.6

Figure X-I. Electron levels prior to application of tension.

X Electroluminescent organicdiodes

Figure X-2. Electron levelsafter contact made (Vapplied

261

= 0).

We should note that a constant potential gradient corresponds to a homogeneous distribution of charges within a volume and that band curvature occurs, typically, in solids in which the concentration of carriers is higher than 10 17 cm- 3 [lei 98, p. 855], and that occurs only for organic solids when they are doped, something not generally applicable to electroluminescent diodes. ~ Electron levels following polarisation On making contact with electrodes and applying a tension of the flat bands, i.e. VBP = (WITO - WCa)/q, we arrive at the situation detailed in Figure X-3 in which the tension VBP = [VITO - VCa] > 0 is applied at ITO and the potential of Ca remains unchanged. Here, the bands are in a rigid conformation and are horizontal after application of the flat band tension VBP (thick, grey lines show the band structure of Alq3 with HOMO and LUMO levels parallel to metal (ITO and Ca) bands and the initial vacuum level) . Once a flat band tension (VBP) and an additional positive tension (Vps) is applied to the anode (ITO), that is Vappl = VBP + Vps, the electron levels take up the configuration described in Figure X-4 in which is represented the resulting band movement.

y Recapitulation of applied schemes (Figure X-5) generally fabricated as detailed in Figure X-5-a .

In practical terms, OLEDs are

Figure X·3. Regimeof flat bands.

262

Optoelectronics of molecules and polymers

Figure X·4. Band schemefollowing polarisation Vappl

= VBP + Vps.

n/V _..VII "\

-"~:-;V~~]-_-.- - -=-:" " - " - 1 =3.3 eV

.z.:fJ ~_ ....

\VITO=

4.6eV

...J '~"" ~

i3 .......

Figure X-5. Practical ITOjAlq3jCa diode: (a) conception; and (b) band scheme.

Given the results shown in Figure X-4, it was possible to prepare the electronic structure shown in Figure X-5-b for the example system. We can see that the drop in tension which appears ofits own accord is in fact what we called VPS (the 'positive supplementary' tension applied to the ITO anode) , and is equal to the total applied tension minus the flat band tension (VFB). The latter is such that VBP = (WITO - WCa)/q. In this example VBP ~ 2 volts, a non-negligible value in contrast to those generally used of the order of 0 to 15 V. Important information can be gained with respect to the interfaces, where the barrier at the anode, seen by holes , is equal to !1A = IpAlq3 - WITO, which in this example is !1A = 1.3 eY. Given the widely accepted value WITO = 4.9 eV, !1Acan be considered to approximate to I eY. Following on from the points noted above at the beginning of this Section, we can imagine that the contact at the anode is not ohmic and risks even being limited by Fowler-Nordheim emissions. We can note though that at the cathode, there is no barrier to the injection of electrons (the drop in potential energy for electrons being !1c = XAlq3 - WCa which

X Electroluminescent organic diodes

263

works out as ~c = O.4eY) . Indeed the contact can be assumed to ohmic as in the region of this electrode the Fermi level penetrates the degenerated Alq3 LUMO .

b Experimental behaviour and first interpretations Studies ofI(Y) characteristics have shown that Schottky emissions can give an ideality factor (n) of the order 20, a very high value indeed, and an unusually high diode series resistance at 50 kQ. However, the emission law for tunnelling effects is not completely verified, even at high tens ions, and contrary to theory it has been experimentally shown to increa se with temperature. In fact, only the TCL law is followed completely within domains which yield electroluminescence (see Figure X-6 for the ITO j Alq3 j Ca structure). The same conclusions as those detail ed just above have been reached, qualita tively, for more complicated system s based on ITO jTPDjAlq3 jMg-Ag by Burrows et al. [bur 96]. TPD is a material used to reinforce hole injection, and Mg-Ag as the cathode has, like calcium, a low workfunction, but exhibit s a slightly lower reactivity [rot 96]. For Burrows et al., interface reactions between the low workfunction electrodes and the electron transport layer (EL) introduce states into the HOMO-LUMO gap and result in ohmic injection at the point of contact. On replacing Mg-Ag with In, which has a higher workfunction, the TCL law is no longer followed and current is not volume limited but controlled by the electrode inject ing electrons. More precisely, the ohmic law is followed at low tension s (due to thermally generated carriers of density no) in thick films, while the SCL law is followed in the same regime for thin films (current varies as y 2 j d3, so when d is large, the current is dominated by the simple ohmic law) . As the tension increases, the Fermi pseudo -level rises and traps start to fill up. This decrease in the number of empty traps, which now receive few electrons, induces a rapid increase in mobilities and the overall current, resulting in the TCL law being observed (I ex ym+ I j d 2m+ I) . At a sufficiently high value of V, all traps are filled and we should once again find the classic SCL law, although this type of value of V is rare ly obtained without destroying the films.

(a) (b )

.018 .014

~

80

...c

60

.~ III

.01

-

...

E

'Q

oS

~

... '"c .s'" c 3

40 20 .002

o

o

5

10

15

20

0

0

V o ltuge ( V)

5

10

15

20

Figure X·6. (a) leV) characteristics of the ITO/Alq3 /Ca structure; (b) Luminescence curve with tension for the ITO/ALq3/Ca structure.

264

Optoelectronics of molecule s and polymers

There have been numerous arguments developed as to why the TCL law is observed [bur 94] and [bur 96]: • • •

•

when m is sufficiently high , I <X V Id 2 , so that when I is a con stant , V = f(d 2 ) is a straight line as indicated in Figure X-7-a; as T, = Et /k = mT ,mmustincreaseasTdecreases,again showninFigureX-7-b; additionally, m = Tt(l/T) , and plotting m = f(l/T ) gives T, = 1780K and Et = 0.15 eV which is a value in agreement with that given by quantum calculations (0.21 eV and detailed in Chapter IV, Section III-3 (see also Figure X-7-c)); taking the law I <X Vm + 1Id 2m+ l , it is possible to determine N, = 3 x 1018 cm- 3 and N, = 1019 cm- 3 (also using the formula for Jexp given in Chapter VI, Section IV-6-b). In addition, from the volume mass, the density of molecular sites can be evaluated to the order of 2 x 1021 cm- 3. We can conclude that electrons injected into LUMO levels are subsequently strongly localised and with a high proportion of which being held in trap levels, from which they can recombine (radiatively) with holes; nevertheless, their low concentration with respect to the overall number of molecular sites can explain their low quantum yield in Alq3 (around I %). Figure X-7-d gives a general view of the mechanism of electroluminescence.

(b) I (A)

10.3 (a) ~

10"

iZO

05

10

c

10-6

~c 15

07

10

~

10-1 09 10

010

m

=

10

10011

( d) ~ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - i

: I

• Franck • Condon

...In

,I, L...:....:..:= Il ol, InJm lon

(e) m

15

10

and dlfTlLu on :

t..- 40OI.,.

..:<;..""T'!'-#"""'-"---"'''"'----....:j /

E•.e'

..-.-

..

10

Electro n injection and TeL current Op tical ga p: 2.7eV

: I

:

: : I

a

Recofhblnatlon zone ~ - - - -------- - - - - - - - - - -- -- - - -- - - - --- ----- - ---

Figure X-7. (a) Line V = f (d2 ) in which d

== film thickne ss. (b) I(V) curves with varying T. (c) Line m = f(l /T) in which the slope gives T" that is Et = kT t . (d) Electroluminescence mechanisms in Alq3.

X Electroluminescent organic diodes 265 An 'academic' hypothesis ofthe TCL model operating with the trapping time being longer than the average transit time (for a hop between two molecules in the LUMO) was proposed to answer criticisms concerning electronic transport in the model, which envisages a band structure arising from the solid (see discussion in Chapter VI, Section IV-2-a). The mechanism of transport limited by traps (TCL) simply demands a sufficient delocalisation of charges to establish a thermal equilibrium between free and trapped charges [bur 96]. As we have seen, N, ~ Ne , indicating that each electron introduced into the LUMO band will subsequently and automatically trapped on an Alq3 molecule . A relaxation would follow yielding a polaronic state which would then result in conduction occurring through hopping mechanisms appropri ate to these quasi-particles (with reduced mobility, as experimentally confirmed with Itn ~ 5 x 10-5 cm 2 V- 1 s"),

c Controversial points and proposed improvements [ioa 98] a Controversial points [ioa 98] under discussion: • •

There are essentially two details which remain

that concerning the actual existence of the band scheme for small molecules , although this argument is not discussed here; and that which relates to the hypothesis of invariance of mobility with respect to an electric field, which is used in establishing the SCI and TCL laws. There are the following considerations which can be brought to mind: - up until now, because of the low mobilities and short mean free pathways (mfp) observed in organic solids, it was thought that between any 2 collisions the carriers could not gain much energy from an applied field, and that mobilities and transport were governed by temperature (T) through thermal vibrations overcoming barriers and were essentially independent of the applied field (in contrast to inorganics) . - However, in disordered systems, the general expression for It is empirically obtained and has only been verified through Monte Carlo simulations in terms of position and energy for individual jumps between disordered sites [bas 93]: It

= Ito exp (~2~~) exp ( C {[k;

r-}:2} E~/2)

in which Ito is the mobility in a zero field, a and}: are parameters which characterise, respectively , the energetic (diagonal disorder) and positional (non-diagonal disorder) degrees of disorder [cam 98b] and [shi 98], while a represents the width of the Gaussian distribution of transport states and t}. = 2a/3 . Alternatively, Gill's empirical law for charge transfer, established using amorphous complexes is also used [gil 72]: It(T, Ea ) = Itoexp

A),

(-~) exp (G kT kT eff

eff

where

T eff

.!.T -To

266

Optoelectronics of molecules and polymers

in which ~o is the activation energy at zero field, To is the temperature at which straight lines J.1 = f (I IT ) with respect to the applied electric field gives an intersection (experimentally, these lines converge at l iTo) . Generally for small molecules, the derived mobility of carriers increases with the applied field. In the presence of energetic disorder, the electric field aids electrons overcome potential barriers due to energy differences between sites, which affect mobilities [ioa 98]. The mobility follows a Poole-Frenkel like law (which works over concentration-see Chapter YI, Section YI) of the form:

from which vananons with temperature can be obtained by placing 1: = 1:oexp(-~ /kT) into J.1 = qt / m. However, in certain materials such as polymers doped with molecules to the order of 5 to 25% by mass, mobilities do decrease above a certain value of the applied field Ea. As shown in Figure X-8, in qualitative terms, we can suppose that this effect is due to 'diagonal disorder ' associated with the variable distances between molecules [gre 95]. Under a weak field, there are many available and facile pathways (short distance hops in different directions with respect to the field) which are, nevertheless, removed once a strong field is applied . ~ Possible models The presence of trapping levels has been shown not to be necessary by Conwell's group, by modelling the characteristics of organic materials [ioa 98]. The group has even supposed that such levels do not exist, because in examples where only electrons have been injected (using the same LiF/AI contacts on either side, which are supposedly ohmic), the I(Y) curves are identical whether or not the tension is interrupted during the measurements, indicating negligible electron trapping. We should consider, however, the kinetics of such an experiment-see the preceding Section II-I-b. Experimental characteristics may be verified simply by using Ohm's law, into which can be included the appropriate law for variations in mobility: J.1 = J.1o exp(aE~ /2) where J.1o is the mobility under a weak field and a a parameter increasing with disorder. This has been found to be true when using 1= ne ~Lo[exp(aE~/2)]Ea, with a = 1.3 x 10- 2 cmo.s y- o.s and J.1o = 6.5 x 1O- lOcm 2 y - 1 s-l).

Electron trajectory under weak field: collisions are

disorientated and electron folio"... energetically favourable path, Transport ilio not dependent on pecitional disorder,

Under a strong field, collision: arcorientated andelectronh3..~ enforced trajectory . On meeting a barrier. electron passage is arrested nd mohilit',' decreases when r~ is verv hi2h.

E : siro n!: field

'IJf\

~

E

A

weak field

+

Figure X-S. Qualitative description of charge transport in strong and weak fields.

X Electroluminescent organic diodes

267

Its important to state though that the presence oftrapping levels has been confirmed using a variety of techn iques: •

•

Thermo-Stimul ated Luminescence (TSL ) [for 98] in which the principal peak was modelled using a distribution of traps with energies between 0.25 eY and 0.15 eY. Two supplementary peak s were obse rved toward s lower energ ies (around 0.07 eY); and Photodipolar absorption [jeo 0 Ib1 which results from a thermo -photo -die lectric effect due to: - a photonic, preliminary luminescence exci tation with elec tron ' pumping' onto trap levels (typica lly with a Wood lap emitti ng light at ). = 365 nm); - a thermal effect with an initial cooling then heating of the sample; and - a dielectric effect by reorientation of dipoles associated with trapped charges using a radio frequency field.

Thi s technique has given rise to traps with depth s of approx imate ly Et = 0.2 eY, and has shown that trap s can persist, following excitation with daylight , for around 1 hour. It seems reaso nable eno ugh to assume that effec ts due to space charges and mobil ity dependencies on the applied electric field should affect J(Y) charac teristics [she 98]. However, if we look close ly at these charac teristics with respec t to TCL representations (which is the most wide ly respec ted law), then we can see that they do not follow perfectl y straight lines. There is very possi bly an improve ment which can be made to the TCL law by taking into accou nt mobil ity dependence on other parameters such as tempera ture and the electric field. If we look again at the general expre ssion for current density, ignoring the diffusion term, we have J = pv = qnu.E , that is, J = qn~ Y/ d. In general, as above disc ussed , the concentratio n n of current limited by space charge are in the form n ex V'" . Once only free (untrapped) charges contribute to the space charge, then m = 1 (giving the saturation current J, in Mort-Gurney' s law). If charge space is dominated, however, by trapped charges (as can be supposed given the above detai led results) within traps of expo nential energy distribution, we now have m = Tci T where E, = kT c . As detailed above, we now have J ex y rn + 1, and in addition, if we suppose that mobility (u) is not actually a constant, the J(Y) law will depend on variatio ns in I-l and will no longer follow a law based on y rn + 1, if ~ varies with the electric field, that is with Y/ d (where there is a break with the TCL law). The esse ntial problem is to reso lve the dom inant effec t (space charge or mobil ity) in any specific tension domain . If we can suppose that mobi lity varies negligibly with respect to E a , so that Laplace's equation may be integrated, we can direc tly introduce the appropriate mobil ity law I-l (Ea , T) into TCL type equations.

3 Polymers a Initial models [Par 94] Emissions through tunnelling effec ts (Fowler-Nordheim) In the ca se of poly (2methoxy,5-(2' -ethylhexoxy )-para-phenylene vinylene), commonly called MEH-PPY,

Ct

268

Optoelectronics of molecules and polymers

Parker showed by experiment that carrier injection is controlled by the height of the electrode-polymer barrier. For the demonstration, diodes which gave preferential injection of one charge carrier type over the other (see Figure X-9) were used. This effect is classically obtained by varying differences between the workfunction of the cathode metal and the electron affinity of the polymer on the one hand, and the workfunction of the anode metal and the ionisation energy of the polymer on the other. In so doing, at the metal-polymer interfaces, one can favour the passage of holes and destroy that of the electrons, as shown in Figure X-9-a, or indeed favour the passage of holes and destroy that of the holes , as in Figure X-9-b. With MEH-PPV, the best results have been obtained using a calcium cathode and an indium tin oxide (ITO) anode. Experimental curves have shown that the I = f(V) curves: • •

uniquely depend upon the value of the electric field (and not the tensions, as is found for Schottky inorganic structures); and are virtually independent of temperature.

This behaviour can be seen as characteristic of tunnelling effect, following the FowlerNordheim equation, over a triangular barrier. Log(I/E;) = f(I/E a ) is a straight line, and permits a deduction of the barrier heights at the interface between the injecting electrode and the polymer (for ITO see Figure X-9-a, and for Ca see Figure X-9-b). This conforms with the scheme for rigid bands as the bands curves can only be at a minimum given the low carrier density within the polymer, estimated to be of the order of 10 14 cm- 3 from C(V) measurements. ~ Functioning voltage and yield Generally speaking, I-V characteristics are essentially determined by the majority carrier, which are in turn determined, be they holes or electrons, by the lowest electrode-polymer barrier, respectively, situated between the anode and the polymer or between the cathode and the polymer. However, the yield of a diode is determined by the minority carriers injected at the highest level of the barrier, a parameter which we will look at in more detail below. Indeed, the

Vacuum level 2.8eV

Ca

1-::=== f-

4.geV

Anode (+)

Organic film (a)

In Al

f----C

A

Cathode (-)

Anode (+) Organic film Cathode (-) (b)

Figure X·9. PLED structure favouring: (a) hole; and (b) electron injection.

X Electroluminescent organic diodes

269

yield can be varied by changing the barrier observed by minority carriers, and this can be done without modifying the function voltage (the barrier observed by majority carriers). The inverse is also possible. On going from ITO to Ag and then to Ca, as shown in Figure x-tO-a, the anode tension must be increased so that hole injection can be maintained (respectively, 10, 15 and 25 V). The slopes of the bands ensure that tunnelling effects are as large as the workfunction of the metals are low. There is an optimised yield when the barrier observed by minority carriers is zero , as in Figure X-tO-b, i.e. the barrier at the cathode is as low as possible. However, the lowest functioning voltage is obtained when a zero (minimum) barrier is observed by the majority carriers (at the anode) . Diode optimisation is generally accepted when the chosen electrodes are such that the workfunction of the anode is equal to the ionisation potential of the polymer, and that the workfunction of the cathode is equal to the electron affinity of the polymer. For the system here, using MEH-PPV, the best pair of electrodes are thus ITO for the anode and calcium for the cathode (See Annex A-II, Section II-I for the description ofMEH-PPV). The yield is around I% for a field of approximately 3 x t0 7 Vm - 1• We can remark that the barrier observed by holes is around 0.2 eV, and for electrons is near 0.1 eV, and that the role s of majority and minority carriers has been reversed. It has been shown the yield can be improved when using Ag as an anode, and at the cathode, with Sm (2.7 eV) or Yb (2.6 eV), the operational voltage does not decrease and chemical reactions between the low workfunction metals and the polymer may occur-risking the formation of interfacial barriers. y Starting andfunction tensions The starting voltage depends only on the band gap of the polymer, and would be equal to this gap if it were not for non-zero barriers at the interfaces (see Figure X-II-b). The tension does not depend on the thickness of the polymer film-as verified by the characteristics shown in Figure X-II-d. The minimum voltage required for the diode to function-that is to produce tunnel effects and emit a visible light-as shown in Figures X-II-c and d, does however depend on film thickness, as the tunnel effect is a current fixed by the field, which has a value

Figure X-IO.(a) Effectson tunnellingat the barrierdue to workfunctions; and (b) an optimised component.

270

Optoelectronics of molecules and polymer s

PP V

Ca

PI' /

" 2..J' cV ITO /

eVappl

ITO ------~~ . _ - - - - - -

(a ) Zero bias (V,\ l' = 0); no tu nneling,

(h) Flat hand condition: lInsl'l mltllgl' for tunneling to uccur; eVrll=\Vn o • \" eo = 1.11 -v ( = El:l'r, ' if harrier at electrodes is zero ),

------

(cjf'or ward bias: Tunnelling injection uf both carrier types,

IlIgl~47S ,

, ,,

00 , 1200 (1I)

Ohmi c zo ne

L:!::::========~_L8JiL..-_-.llIg V Figure X-ll. Band structures ofITO / PPV / Casystem: (a) zero bias; (b) at flat band condition; (c) forward bias; and (d) corresponding log I = f(logV) characteristics.

dependent on film thickness. The minimum functional voltage is sensitive to barriers at the electrodes, in contrast to the starting voltage which would vary directly by ,OJ eV for a barrier of 0.1 eV, as detailed in Figure X-II-b.

b The roles of current limited by space charges (SCL), traps (TCL) and double injection (VCC) Bradley and his team have shown that for PPV used in the simple set-up ITO IPPV I Al [Cam 97], the Fowler-Nordheim law did not correctly account for experimentally obtained results with respect to temperature, film thickness and the amplitude of I(V) characteristics. At high tensions, the law observed was of the TCL type , with an exponential distribution of traps of average depth of 0.15 eV. At lower tensions, the law followed was space charge limited (SCL) , exhibiting a mobility which varied with temperature and field in accordance with the Arrhenius law. The transport through the volume can be seen as a hopping mechanism between states distributed following a Gaussian curve , with deep sites playing the role of traps. Another study, run simultaneously to that detailed above and following the same idea as that detailed previously [par 94], was carried out in the Philip s laboratories [Blo 97a , Blo 97b and Blo 98]. The Fowler-Nordheim law gave rise to currents considerably higher (by several orders) than those found experimentally. Two mechanisms were studied, those of hole and electron transport. Here we will describe each individually.

X Electroluminescent organicdiodes

271

a Hole transport Using the ITO/PPY/AI structure, which favours hole injection only, it was shown that current followed the SCL law, as shown in Figure X-12-a. We have Is = (9/8)£l-lp y 2 /d 3 , showing that current is volume and not electrode limited. With s = 3, hole mobility could be estimated as I-lp = 5 10- 11m 2y- 1s-l . It should be noted thatat high fields, a deviation from the SCL law occurred (experimentally, the current was higher than that given by Js, as shown in Figure X-12-b). As detailed above, because of 'non-diagonal' disorder associated with variable distances between molecules, under low fields facile transport pathways are available (short hops in directions at variance with the applied field) which become inaccessible at higher fields. As the field strength increases, the mobility follow s a Poole-Frenkel type law (assuming that traps are not responsible for disorder, as there is a quadratic SCL regime without traps unlike the TCL system). With I-lo the mobility under zero field and /::". the energy of activation, I-l = I-lo exp( -/::". /kTo) exp(aE 1j 2) . This gives /::". = 0.48 eY and I-lo = 3.5 x \0-3 m 2 y-I s-I. The factor a has been empirically defined with respect to temperature as a = G

(k~

- k+0)' The insertion of I-l into Js has been used to obtain

an excellent agreement with experimentally obtained l(Y) curves . Here, G = 2.7 X \0-5 eYYo,s m-o,s and To = 600 K, values similartothose obtained forpoly(N-vinyl carbazole) (PYK). We should note again , however, that to obtain Js from the SCL law, we have to suppose that I-l is a constant with respect to x, but E = E(x), so we have to state that the variation is sufficiently small so as to be able to perform the integration!

B Electron transport The example used here is that of Ca/PPY / Ca which favours electron transport and is shown in Figure X-13 . The current due only to electrons is less than that due only to holes, up to 3 orders at low tensions , Limited by traps, electron current is strongly field dependent, and follows a TCL law, that is J ex: yl+ l / d 21+ 1, in which traps are exponentially distributed, The slope of log J = f (log Y) gives m and T, = 1500 K and N, = 10 18 cm - 3 . y Recombinations On the basis of bimolecular recombinations, mechanisms involved in recombinations have also been studied [Blo 97a] . The I(Y) characteristics

r'

E

P'

s..,

E

..,s

0.1

10 V (V)

0.1

10 V ( )

Figure X-12. (a) SCL I(V) curves at Tambient ; and (b) deviation from the SCL law at low temperature and under a high field,

272

Optoelectronics of molecules and polymers

~::c/~/ /4m 221m

10'"

10

100 V (V )

Figure X-l3. I(V ) characteristics for electron transport. give rise to only a slight difference bet ween single hole and double injections. Double injections are hole dominated while electrons are highl y trapped. The recombination con stant has been estimated to show that only around 5 % are actually radiative, the greater part in PLEDs occurring non-radiatively.

III Strategies for improving organic LEDs and yields 1 Scheme of above detailed proces ses Figure X- 14 detail s the vario us stages towards electroluminescenc e [Bra 96]. To give a summary of the different stages, covered above: •

A in which currents {I} (elec trons) and {2}(holes) are injected, respectively, giving rise to leak currents {I' } and {2 ' }. Figure X- 14 shows curre nt densities and carrier speeds;

[Cd Materiulair interface [U)

Figure X-14. Basic mechanisms in electroluminescence.

X Electroluminescent organic diodes 273 • •

•

B in which electron-hole recombinations occur, resulting in the initial formation of excitons in singlet {3) and triplet {4) states ; C in which excitons are dissipated either through the radiative {6) (singlet ~ singlet) or non-radiative (7) (singlet ~ extinction centre) and {8} (triplet ~ singlet) pathways; and D in which there is refraction of the light beam on going from the material to air interface.

2 Different types of yields As detailed in Chapter VIII, in general terms, different types of yields may be defined as:

a The external quantum yield This yield, the external quantum yield, is defined by: Il ext

=

number of photons emitted outside the structure . . = T]EL· number of Injected charges

The external quantum yield can also be thought of as the product of two other yields: T]ext = T]opt T]<jlint. where T]<jlint is the internal quantum yield and Ilopt the coefficient given in Chapter VIII to account for refractions at the diode-air interface (step D in Figure X-14) . With an organic material with an indice n greater than that of air, only internal emissions incident to the interface at an angle below the limiting angle 8\ can escape towards the exterior. The result is a non-negligible loss, as the factor T]opt ~ 1/2n 2 (calculated in Chapter VIII, Section 1II-2, using Lambert emissions) is of the order of 0.2 when n = 1.6. External yields are therefore only around 20 % of internal yields . Efforts required to improve this figure are detailed later on.

b The internal quantum yield The internal quantum yield can be thought of as the product of three different factors . Each one is with respect to the steps A, Band C, detailed in Figure X-14, and can be singled out in the defining equation for this yield, T]<jlint = YT]r
•

Y is the level of recombination of injected carriers i.e. Y = Jrlh. In Step A, with JT = Jh + J~ = I, + J~ (total current density) and Jr = Jh - J~ = Jc - J~ (recombination current density) then Y = ) if there arc no leak currents a~ = J~ = 0) and there is an exact equilibrium between the two types of current, i.e. Je = Jh. However, for example, if all holes are used up in recombinations, J~ = 0, then some electrons will traverse the whole structure without recombining giving J~ =1= 0, and Y < I; T]r is the generated fraction of radiative singlet excitons, as denoted in Step B, due to the probabilities of the spins required to yield electroluminescence from

274

•

Optoelectronics of molecules and polymers singlet states (S = 0, Ms = 0) and triplets (S = 0, Ms = -1 , 0, 1). 11r can be directly estimated from 11r = us /[Us + vrl = 0.25 = 25 %; and <1>[ is the fluorescence quantum yield (see Step C) and while always below I, it can reach up to 0.7, the loss being due to inevitable, non-radiative recombinations {7}. The latter are due to singlet excitons recombining, in general, close to the interfaces, or because of the presence of extinction centres such as impurities and non-radiative traps . In addition, bimolecular recombinations do not operate at 100%, even with the low mobility of carriers in organic media.

c The quantum energy yield The quantum energy yield is expressed in W W- 1 and represents the ratio Emitted luminous energy

hv = 11EL- Supplied electrical energy - nchargeseV eV

11 -

-

nphotonshv

From the above equation it is evident that for a given emitted wavelength (given hv), the lower the operational voltage, the higher the energetic yield. This explains the interest in trying to reduce the operating voltage . Typically, organic materials exhibit values of hv/ev of around 0.3.

d The luminous yield The luminous yield takes into account the photo-optical respon se of the human eye, which is defined by the relationship 11L = 11 . K(A.) in which K(A.) is constant relating observed and actual scales of energy.At A. = 555 nm, K(A.) = 683lm W- I . This yield is thus expressed in lumen watt" I .

3 Various possible strategies to improve organic LED performances Luminance, which depends on the intensity of emitted light , is determined principally by the current density of minority charge carriers and then by the number of these carriers which undergo radiat ive recombinations with the majority carriers. The quantum yield , which can be written as the product of the four terms detailed just above 11EL = 11 opt 11<1> int = 11 opt Y11 r
the highest number of minority carriers undergoing radiative recombinations. This in turn necessitates the use of efficient electrodes and injection layers for these carriers (with a high injection current or ohmic contact). Given that interfaces tend to favour the presence of defaults, which act as extinction centres of luminance, it is judicious to try and displace the recombination zone for the minority carriers towards the bulk of the material. This necessitates the use of a minority carrier transport layer, as the injection layer may not necessarily be good for minority charge transport ;

X Electroluminescent organic diodes

•

•

•

•

275

an improved method for allowing otherwise trapped emitted light to exit the device . Microcavities, structures which diffu se light , can help direct the emi ssions and impro ve the otherwise highly detrimental value of lIopt; as high a possible level of injected carrier recombination . y = Jr/Jr will increase if: - Jr is large and , as previously noted , there is as great a current of minority carriers as possible, which in turn demands as Iow a leak current as possible. The use of layers confining the hole s and electrons can respond to this demand; - Jr is small and therefore the majority current is not too great with respect to the minority current. The injection of majorit y charges which do not go on to recombine with minority charges adds nothing beneficial to the optical behaviour of the device, uselessl y consume s current and results in a harmful heatin g effect which reduces its lifetime [tes 98]; the highest possible level of radiative exciton production possible (i .e. a high value of lIr). This in turn means recovering the highest possible percentage of triplet excitons which result in electron injection. The use of phosphorescent materials and mechanisms which can transfer triplet to singlet states can be envisaged; and the fluorescence quantum yield (f) should be as high as possible. Generally, in the solvated state , this factor can be very high , even near to I (for example, laser dyes). However, in the solid state, this value is generally much lower, typically of the order of 0.1 to 0.5. Thi s is due to forbidd en dipolar transition s between the fundamental and lowest excited states which appe ar, either localised on a dimer or as a band in the solid state, following the breakdown of degenerate levels present in the solvated state [sch 00] . Thu s well organised and crys talline states should be avoided, as we have already see n in Chapter VII, Section VI-6. There is nevertheless a condition that can be imposed upon the latter remark, and that is that any disorder impo sed should not be so great as to limit any gains in transport (injection and that of minority charge carriers) which can arise from the organi sation of a system (where charge mobility can be high) and would otherwise impro ve the fluorescenc e yield . Thi s is a characteristic of certain types of material s, for example those based on discotic molecule s [seg 01]. In addition, as previously mentioned, non-radiative centr es should be excluded, by the use of layers free of structural faults or other imperfe ction s.

Another, delicate problem is that of ageing, which results from external effects or from the electrodes. The former may be resolved using encapsulation, while the latter is slightly more difficult but can be resolved using barrier layers, an example of which is PPX deposited via VDP (see Chapter VII, Section 1-3). There is no way we could even pretend to detail all the possible strategies that have been devised to answer the above criteria. Instead , we shall limit this Chapter to only the most currently used solutions, especially when we look at actual appl ications of these devices. Nevertheless, we shall briefly detail why some organi c solids are p-type (polymers such as PPV or poly(pa ra-phenylene) (PPP» and why others are n-type (small molecules such as chelates), at least when protected from oxyg en.

276

Optoelectronics of molecules and polymers

IV Adjusting electronic properties of organic solids for electroluminescent applications 1 A brief justification of n- and p-type organic conductivity a Small molecules The best known example is Alq3, in which the chelation, or complexation, of the AI3+ ion with 3 ligands leaves the aluminium reduced in electron density to become effectively an electron acceptor. When assembled with other Alq3s, this character results in a facile transport of electrons, and thus Alq3 is considered an n-type material. Another example is that of layers of the complex rare earth complex of diphtalocyanine PC2M, where Pc is [C32HI6NS]2- and M is the rare earth ion. These materials are intrinsically n-type (under vacuum) . Scandium diphtalocyanine is paramagnetic, as its outer orbital is but half-filled and has unpaired electrons delocalised over the phtalocyanine macrocycles. The n-type conductivity, under vacuum, can be due to intermolecular transfer of these single electrons. On contact though with oxygen, the gas is cherni-absorbed and forms oxygen- PC2M bonds . The oxygen cannot be removed by the simple application of a vacuum, due to the formation of a charge transfer complex in which the PC2SC is lacking in electrons and the oxygen gains a negative charge . The positive state of the PC2SC moves as a hole through diphtalocyanine molecules which surround the negatively ionised oxygen molecule. In effect, these materials change to being p-type once exposed to air, as do many polymers which are subject to a similar mechanism.

b Polymers In general, polymers tend to behave as p-type semiconductors. The origin of this can again be contributed to the presence of oxygen [gre 94, p 58] . We can also add that in the case of PPV, obtained like many other polymers from the thermal conversion of a precursor, that the process used for its preparation can give rise to many pendent bonds. A possible result is therefore the transfer of electrons through the corresponding levels, also possibly generated by thermally diffused oxygen, from levels at the highest point of the polymer valence band . The latter thus generates holes. The generation of this p-type character of polymers can also be evidenced on doping using ion implantation [mol 96].

c Adjusting the type of conduction The type of conduction of a material can be changed by altering the 'backbone' of the organic material, simply by adding on electron donating or accepting groups (see also Chapter XII, Section 11-3). For example, PPV can be modified to facilitate : •

the injection of electrons by increasing the electron affinity (X) of the polymer on adding electron attracting groups, such as cyano (CN), onto its backbone.

X Electroluminescent organic diodes

277

Figure X-IS. Structure of cyanated poly(para-phenylene vinylene) (eN-PPV).

•

Figure X-IS shows the example of eN-PPY [bra 96]. Even aluminium can be used to replace calcium as the cathode with this material, without altering the yield; the injection of holes by stabilising their presence on the polymer by modifying it with electron donors such as alkoxy (-OR) or amine (-NH2) groups.

2 The problem of equilibrating electron and hole injection currents As detailed above, the equilibration of electron and hole injection currents is essential in order to generate excitons. If this cannot be done then: •

•

a large current of electrons or holes can pass through the device without generating excitons, an obligatory step in the process of light emission, thus reducing the yield; and there is an initial formation of excitons only to be reduced in the neighbourhood of one or the other of the electrodes (the cathode or the anode if, respectively the majority current is holes or electrons) in a process tied to the presence of default extinction sites of the organic-inorganic interface. There are two methods by which an equilibrium can be reached (Figure X-16) :

•

•

adjust the workfunctions of the metallic electrodes so that the barriers at both organic-inorganic interfaces are close, in a method only valid if the current is determined by injection and not by the bulk. However, this can mean the use of reactive electrodes, which suffer from problems which will be detailed in Section 3 below ; and modify the organic solid to the workfunction of the metallic electrodes so as to reduce the size of the barriers at the two interfaces. Electron affinity and ionisation potentials are parameters used to choose appropriate materials, although the choice is also governed by the type of transport.

3 Choosing materials for electrodes and problems encountered with interfaces A very simple and conventional structure is illustrated in Figure X-16, into which PPY or Alq3 can be placed .

278 Optoelectronics of molecules and polymers ......................... _.-

.......»:

--_._.._._..._. ·.. .t

....

Ca

IT

Figure X-16.Modification of electrode and bulk materials in a simple system.

a At the cathode Briefly : •

•

Metal from the cathode can diffuse into the polymer, which becomes doped. The injection barrier is therefore moved to between the doped and undoped parts of the polymer. Covalent polymer-metal (aluminium) bond s can develop with the generation of a barrie r toward s charge injection at this level; Different alkaline-earth metals have been used with Alq3 . With Ca, it is apparent that the presence of residual oxygen (10- 6 mbar durin g deposition ) forms a passivation layer which arrests interactions (Ca with N then a of Alq3 if Ca is deposited on Alq3), or degradations of Alq3 (if Alq3 is deposited on Ca). With Mg as the cathode, oxygen needs to be completely removed. However, the injection barrier at the interface of Al and Alq3 is actuall y reduced if a thin layer (0.5 nm) of LiF (or CsF) is introduced between Al and Alq3 . The introducti on of a thin layer ( I nm) of MgF 2 between Alq3 and an alloy of Mg:Ag ( 10: I) increa ses both the quantum yield and lifetime.

b At the anode The most typically used substance for the anode is mixed indium and tin oxide (ITO). It has a square resistance between 4 and 80Q /D. An inconvenience which does arise with this material is its ability to generate oxygen, which can diffuse toward s the organic solid and result in a degradation due to photo-oxidation. The p-type character of conjug ated polymer s makes them good candid ates as material s for anodes, favourin g the injection of holes. They can also play a role in protecting against oxygen . Interesting effect s were observed with doped polyaniline, which being opticall y transparent was used to replace, or used on, ITO. As we shall see, in multi-la yer structures, polyethylenedioxythiophene (PEDOT) can also play the role of barrier against diffusing oxygen and hole injection if (J > 2 Q-I ern" I , as can copper dipht alocyanine .

X Electroluminescent organic diodes

279

•

v + eII.nod< /......... /

////

Confinement zone (exciton

formatlon )

O --f=-~

Exnmple

ITO

PPV

C -PP

Ca

Figure X-I7. Confinement zone formation.

4 Confinement layers and their interest It was very quickly realised that confinement layers were necessary to improve the probabilities of carrier recombinations [fri 92], even though organic materials provide high probabilities due to the relatively low carrier mobilities . The use of this layer though also allowed greater separation of the recombination zone from the electrode interfaces. As shown in Figure X-17, the confinement zone is obtained by choosing materials which at their interfaces form barriers against holes coming from the anode and against electrons coming from the cathode.

V Examples of organic multi-layer structures: improvements in optoelectronic properties 1 Mono-layer structures and the origin of their poor performance As an example of a mono-layer structure we will detail ITO/Alq3 /Ca. Figure X-5 shows that while electrons may be easily injected at the cathode towards the emitting layer, as there is no potential barrier, a similar remark cannot be made for holes . The high potential barrier, around I eV, which holes observe at the interface between anode and Alq3, reduces any charge injection. This can only be overcome by the application of a high potential, which results in charge transfer, notably, by tunnelling effects . As previously discussed, luminance and electroluminescent yields are strongly dependent on the minority current. With the Alq3 molecule being n-type, the difficulty found in injecting holes explain s to a greater degree the poor luminances obtained with mono-layer structures. To reduce these problem s, different types of layers can be used and there are examples of di-Iayer, tri-Iayer, and even multi-layer structures.

280

Optoelectronics of molecules and polymers

2 The nature of supplementary layers Independent of the nature of the emitting organic layer (the organic material used original in a mono-layer device) , various other layers can be introduced. Generally speaking, these layers can be termed: • • • •

Hole Injection Layer (HIL); Hole Transport Layer (HTL); Electron Injection Layer (ElL) ; and Electron Transport Layer (ETL).

The following paragraphs detail the way in which these layer s function and will exclude descriptions of the operation of chemical groups from which they come, a subject covered elsewhere [bar OOb]. Appendix A-I 0 does however give the chemical formulae for molecules and polymers from which the layers are formed . HIL and HTL layers are particularly necessary when the emitting layer is made of a small molecule such as Alq3 which favours electron transport. When the emitting layer favours hole transport, for example with polymers and notably PPV, it seems reasonable that to aid minority carriers, electrons injected at the cathode, ElL and ETL layers should be used. Here , we will study only examples relating to small molecules. It is worth adding that, as we have seen in Chapter VIII, there are effects associated with different physical treatments of the emitting layer, which can be similar to those obtained using various chemical layers . We shall see in Section 4 that the densification by ion beam assisted deposition of the emitting zone in contact with the anode can augment up to 10 times optoelectronic performances of OLEDs.

3 Classic examples of the effects of specific organic layers [tro 01] a HIL layer effect The HIL layer is in contact with the anode . One of its many roles is to perfect the planeness of the interface, reducing effects caused by points which otherwise locally rupture the dielectric field. Another of its roles is to act as a barrier against impurities, notably oxygen, which diffuse from the ITO anode towards the organic emitter. If allowed through , oxygen gives rise to premature ageing of the organic layer and non-radiative centres may be formed . The most widely known material, which fulfils this role, is poly(3,4ethylenedioxythiophene) doped with poly(styrene sulfonate) (PEDOT-PSS) and it can be prepared as a film using spin coating. Alternatives, although less efficient, are available in the form of copper phtalocyanines. In order to study the benefits of using an HIL layer, optimised structures of ITO /AIQ3 /Ca/AI and ITO/PEDOT-PSS /AIQ3 /Ca/Al have been compared. It was found that the monolayer structure had a performance which was considerably improved upon by the bilayer structure. As shown in Figure X-18, the luminance for the former was 49 cd m- 2 (at 18.0 V) while the latter displayed greater than 5000 cd m- 2 (ate .a. 9.2 V). The threshold voltage, in the same order, dropped from 8.0to4.1 V,

X Electroluminescent organic diodes 10000 Luminance (cd rn-

281

z)

1000

~

50 nm PEDOT+ 60 nrn Alq3

-SOnrnAlq3 100

10

Tension (V) 1 +------h------f.~...------.-----___.

o

5

10

20

15

Figure X-IS. L = f(V) characteristics for ITOjAlq3 jCajAI and ITOjPEDOT-PSS jAIQ3 j Caj Al structures.

Eo No electron confinemen t

--- ---------

) •I 3.1 eV

Ia

~ z.s ev

I I I I

4.7eV

I Hole confinement

I 5.2 eV

---------- --------

ITO

I

t

/

~/~ ~

5.66 eV

Ca PEDOT Alq3 (HTL I - - - -- ''I'lIl, ...

2

n

- - PEDOT-PSS

Figure X-19. Band scheme of the ITO/PEDOT-PSS/ AIQ3/Ca structure.

while the quantum yield improved from 0.02 to 0.72 % (luminance yield increased from 0.02 to 0.991m W- 1) . In Figure X-19 it is evident that the band schem e of the bilayer ITO/PEDOT-PSS /Alq3 /Ca/Al structure has the PEDOT-PSS HOMO level lying in between that of Alq3 and the workfunction of the anode AlQ3 ' permitting hole injection into the structure. There is also a confinement of holes at the PEDOT-PSS /AIQ3 interface.

282 Optoelectronics of molecules and polymers

b HTL layer effect This layer is generally deposited in between the HIL and emitting layers. The materials used for this layer are either sublimed small molecules such as N,N' -diphenyl-N,N'-(3-methylphenyl)-I, I' -biphenyl-4,4'-diarnine (TPD) or polymers such as PVK, which may be deposited using spin coating. These materials display relatively high hole mobilities (J.Lp ~ 10- 3 cm 2 V-I s-I) and have a HOMO band quite close to the Fermi level of ITO. In the bilayer structure of ITOjTPDjAlq3jCa, as detailed in Figure X-20, we can see that on the one hand, the intermediary level of the TPD HOMO level aids hole injection into Alq3, while on the other, the LUMO level appears as a sizeable potential barrier at the TPD j Alq3 interface . The latter stops electrons from going from Alq3 to TPD and confining carriers. Thus at the same time the TPD layer generates more holes and more electrons in the Alq3 layer, increasing the number of radiative recombinations and improving the luminance from 49 to 9600 cd m- 2 , as shown in Figure X-21.

c Trilayer structures Individual studies of the effects of HIL and HTL layers on diodes have demonstrated that: • •

the HIL layer allows lower working voltages; and the HTL layer permits confinement of electrons and holes at the HTL/Alq3 interface.

2.2geV

4.7 eV

t ITO

5.37 eV

.. f'.

!e2.g ev

I 3.1 eV I I I I I I

:x f 5.66 eV L~

TPD

Alq3

Ca

Figure X-20.Band scheme for the ITOITPDI Alq3 lea structure.

X Electro luminescent organic diodes

10000

283

Luminance (cd m" )

--tr-

1000

40 nm TPD + 60 nm Alq3

-e- 20 nm PVK + 60 nm Alq3

-

80 nm Alq3

100

10

Tension (V)

1+-----r-----+.......--I---~----"""'T"----

o

10

5

15

20

= f(V) characteristics for ITO/ Alq3/ Ca/ A1, ITO/ TPD/ AlQ3/ Ca/ AI and ITO/PVK/ AlQ3 /Ca/ Al structures.

Figure X-21. L

Figure X-22 -a shows how a structure ca n be made to combine the effec ts of both of these layers. Th e resu lts, shown in Figure X-22-b show that this optimisation can give rise to a luminance of arou nd 22500 cd m- 2 (at 12 V). Th e worki ng voltage actually used can be optimi sed by varying the thickness of the various layers [tro 0 1].

(a)

2.29 eV

Co nfinem nt

~

2.9 eV

TPD

Ca

4.7 eV 5.2 eV

ITO PEDOT

Figure X-22. (a) Band scheme of the ITO/PEDOT-PSS /TPD/ AIQ3/Ca / AI (2) structure, (b) L

= f'(V) characteristics of bi- and tri-Iayer structures.

284

Optoelectronics of molecules and polymers Luminance(cd m-2)

(b) 20000

-e- 40 TPD + 60 Alq3 -e- 50 PEDOT

15000

-

+ 60 Alq3

40 PEDOT + 20 TPD + 60 Alq3

10000

5000

O. . .- -....... ~~iIEii&iiiiiiiiiiiiiiiiiiiiiiilfOl~~::.,.--..:=~

o

5

15

10

20

25

Figure X-22. (Continued)

4 Treatment of the emitting zone in contact with the anode While physico-chemical treatment of the anode (ITO ) using oxygen plasma can lead to an improvement in organic LED performances, we have seen that treatment of the actu al emitting zone can also improve the injection and tran sport of minority carriers lead ing to an increase, by an order of size, of the luminance and yield of a LED . The effect of this treatment is detailed in Chapter VIII , and here it suffices to present a 1.2

ITO/Alq y'Ca --ITOIP30T /Alq y'Ca ITO/Alq ) assisledlAlq)/Ca

~...

.~ 0.8

f. 'I. /

v"".

..

.....

0.6

~

= ':r

]

.. ·V-

I

......

v

.

E

§

---,

----

-

0.4

~I .I /

\

-,

•

]

..

0.2

I

,i

~

o .......-..-- ................ J....-r-::;'--.A 0

10

15

20

25

30

35

Vo lt age ( V)

Figure X-23. Quantum yields for mono-layer, chemical bilayer and physical bilayer structures, in order of increasing yield. .

X Electroluminescent organic diodes

285

comparison of the yields obtained by three different structure s in Figure X-23 . These three structures are: • • •

a single layer ITO/Alq3/Ca structure; a non-optimised 'c hemical bilayer' of ITO/ P30 T/ Alq3/Ca, in which P30T plays the role of the HTL layer, preferentially transportin g holes; and a 'physical bilayer' of ITO/ 25 nm Alq3 depo sited using ion assistance /50 nm Alq3/Ca, where the assisting beam was of helium ions with energy 100 eV and current density j = 50 nA cm- 2 and the period of assisted deposition was optimised at ~t = 100 s.

The advantage of the physical bilayer is that it require s only one single chemical product to be evaporated, however, the technique does necessitate an ion source and cannot be applied to OLED s prepared using vacuum sublimation.

VI Modification of optical properties of organic solids for applications in electroluminescence 1 Adjusting the emitted wavelength There are several ways in which the wavelength of the emitted light can be varied: •

The size of the forbidden band varies with the conjugation length n of a polymer (or oligomer). When n is increased, for example in going from an oligomer to a polymer, there is a global evolution in the number of interacting states and thus a decrease in the size of the forb idden band. The result is a shift in emissions to the red. Inversely, the use of oligomers shifts emissions towards the blue. In the case of PPP, an empirical law has been given detailing the absorpt ion peak (Eo) [lei 98]: 3.16] Eo = [ 3.36 + ~ eV ,

•

•

which explicitly shows how, for example, when n decreases, on using oligophenylenes, Eo increa ses; when monomer units within a polymer gives rise to weak Jt interactions (overlaps), for example between phenyl rings, the resulting gap is large. For example, PPP is a good candidate for blue emissions as its gap (emission peak at 465 nm) is greater than that of PPV (peak at 565 nm). The introducti on of non-conjugated sequences, which dimini sh the degree of conj ugation, has been used with PPVs to induce a blue shift in the emission. MEH-PPV, PPY modified with alkoxy group s (used by CDT, Cambridge) emits at 605 nm (in the orange-yellow); on turnin g to small molecules, we can see that the most widel y used are organometallic comp lexes, for example based on a central metal ion such as beryllium, magnesium, zinc or gallium and an outer set of ligands which act as

286 Optoelectronics of molecules and polymers

R = H: 8-hydroxyquinoline (q) R = CH 3 : 2-me thyl-8-hydroxyquinoline (mq)

Bebq-

lO-hydroxybenzo[h]qu inoline (bq)

Figure X-24. Representations of Alq3 and Bebqj along with the ligands q, mq and b.

•

emitters (as we have seen in the case of Alq3) . The emission wavele ngth for these spec ies is generally towards the green, but can also be modified slightly by using a different metal cation, thus altering bond lengths with the ligan ds. For example, we have }. = 530 nm for Alq3 and}, = 517 urn for Bebqj , as show n in Figure X-24. Blue gree n and even blue colo urs can be reached by displacing the em itting zone towards TPD, for exa mple. The central ion can also be a rare eart h, giving rise to a change in the emitted wavelength. This is discussed further in Section 4. Another strategy consists of grafting small electrolumi nescent molecules onto polymers [gau 96]. Various possi bilities have been explored, using varying concentrations of dye, such as coumari n. Alone, these dyes have a tendency to crystallise , thus loosi ng efficiency. The problem remains though that such structures requ ire high starting tensions and do not appear to be particularly stable.

2 Excitation energy transfer mechanisms in films doped with fluorescent or phosphorescent dyes Studies of host-guest systems, Alq3 (host) - DCM2 (2% guest fluorescent dye) and Alq, (host - PtOEP (8% guest fluorescent dye), have shown in both cases that exci tations are formed directly on the host Alq3 syste m [bal 99] . However, the emissions of the guest syste ms are very strong, and in the forme r syste m less than 1% of emissio ns come from Alq3. This indicates a high level of excitation transfer from the host towards the guest.

X Electroluminescent organic diodes

287

(a)

(b)

---+-== - - ----.~-

~---~ Fig ure X-2S. Transfer of excitation energy from a host molecule (D: Alq3) to a fluorescent dye guest molecule (A: DCM2). (a) Single host ~ singlet dye fluorescent transfer; 10* + 1A ~ 10 + 1A * Both Forster and Dexter transfer possible, but former is more important (b) Triplet host ~ triplet dye fluorescent transfer; 30* + 1A ~ 10 + 3A * Only Dexter transfer possible here; with fluorescent dye, phosphorescence is only slightly possible.

Gi ven the re sults detailed in Chapter VII , Section VI -7 co ncern ing mechanisms of excitation transfer, we can sum marise the po ssible energy transfer proce sse s from the ho st (D for donor) to the guest (A for acceptor) for a fluorescent dye (Figure X-25 ) and for a phosphorescent dye (Fig ure X-26). If the two mechani sm s, Forste r and Dexter

,

(a)

i) lot

'At~

...

lot~

,i 'A*

-ll.. Internal Iconversion I

'ot~

3

t A*t

~

(b)

_jt_ '0

Figure X-26. Transfer of excitation energy from the a host molecule (D: Alq3) to a phosphorescing dye guest molecule (A: PtOEP). (a) Singlet host ~ singlet phosphorescent dye ~ triplet phosphorescent dye transfer; I D* + I A ~ 10 + I A*. Forster and Dexter transfer are both possible, but the former is more probable. The following internal conversion, I A * ~ 3A *, relaxes radiatively as A phosphoresces. (b) Triplet host ~ triplet phosphorescent dye transfer ; 30* + 1A ~ 101 3A *. Only the Dexter transfer is possible between D and the phosphorescent doping agent A, which radiates.

288

Optoelectronics of molecules and polymers

transfers, are allowed in a given system, then the Forster transfer normall y dominates, due to its ability to act over long distances.

3 Circumnavigating selection rules: recuperation of non-radiative triplet excitons a The problem As detailed in Chapter VII (Section V-3-a), following an injection of electron s, three triplets states appear for each singlet state. In fluorescent material s, only emissions from the latter state occur radiatively, and more or less instantan eou sly, in a relaxation to the fundamental state, which is generally also a singlet state. Three quarters of excitons occur in the triplet state, which principally undergo non-radi ative relaxations, and therefore it is evident that for electroluminescent applications these excitons should be recovered so that they can relax radiatively. Phosphore scent materials can be used to do this. On modify ing the (spin) symmetry of a system, the triplet state s disappear slowly through radiative recombinations. It is this slow decrease in luminescence which corre spond to phosphorescence. While admittedly this method is not highl y efficient , the phosphor escence can be improved by the presence of heavy atom s favourin g spin-orbit coupling, mixing triplet and singlet states. However, there are almost no useful phosphorescent organic material s, which have, however, given rise to a vast choice in fluorescent material s.

b Initial, poorly efficient solution s: a phosphorescent layer and Dexter transfers An initial solution to recovering energy associated in electroluminesce nce with triplet excitons is the introduction, in an organic LED, of a supplementary layer containing phosphorescent material [bal 00 and sam 00] . However, as we have mentioned, there are very few organ ic materials which can generate from their triplet exci ted state a significant pho sphor escence. In additi on, the slow declin e of triplet states in the phosphorescent layer leaves open the possibility of a transfer of these triplet states to triplet states in the fluorescent layer (Alq3), which are non-radiative. In practise, in a donor-acceptor system, where the fluorescent acceptor directly dope s the phosphorescent donor materi al, only transfers over short distances can occur. Forster type transfer s are unlikel y; Dexter transfer s occur more readily due to the proximit y of acceptor and donor molecules, and exciton s hop from one molecule to the closest neighb our simply by overlapping molecular orbitals. The transfer of energy from triplet s states (phosphorescent molecule ) to singlet states (fluorescent molecule ) is highly unlikel y, as the mechanism of hoppin g over short distance s conserves symmetry over the donor-acceptor pair, and only transfer from the donor triplet state to the same triplet state on the acceptor can result. Once excitons reach the acce ptor triplet state, in terms of radiation efficien cy, they are lost as the fluorescent dye can only yield a negligible phosphorescence.

X Electroluminescent organic diodes

289

A change in spin orientation, thus assuring a transfer from triplet to singlet states, is only possible if the donor exciton relaxes before reforming at the acceptor, following a random exchange of electrons. This mechanism though remains improbable as the required disassociation of the donor exciton requires a high amount of energy-of the order of I eV in molecular systems.

c Multi-layer systems and Forster transfers As we have already seen, there remains an alternative transfer mechanism. The Forster transfer is associated with long distances, around 5 nm, and does not require contact between molecules, and permits changes in spin of transferring excitons [bal 00]. Chapter VII, Section VI-7-b gives further detail s. In addition, in order to transfer excitons from a host material to an excited singlet state on a fluorescent dye, a phosphorescent and organometallic intermediate is required. Thus, both singlet and triplet states from the host material are transferred to the phosphorescent intermediate, which in turn allows transfers to the singlet states of a fluorescent emitter, which in turn works radiatively in an efficient manner. So that these transfers are possible, the exciton observes a reduction in its energy at each step, much like a ball descending a stairway and as schematised in Figure X-27. The phosphorescent dye, in this configuration, stimulates the transfer of energy from the host material , which acts as donor, to the fluorescent dye acceptor. The transfer is in effect step-wise. However, the concomitant non-radiative Dexter transfer (illustrated by crossed out arrows in Figure X-20), which permits the passage from the triplet state of the phosphorescent intermediate to the triplet state of the fluorescent emitter, should be limited if not completely suppressed. A near total suppression can be accomplished by using an emitting layer itself consisting of alternate layers of phosphorescent intermediate and fluorescent emitter both diluted within a host which ensures the sought Forster energy transfer. Once any

... X- - - - -

-~

'X, ... Triplet :

Sing let

hv

Non 1'Idi. live

Funda menta l

level

.-----..., ,...------, hosphorescent intermediate

Figure X-27. Schematisation of Forster transfer mechanism (full line), where the phosphorescent intermediate stimulates energy transfer from the donor host material to the acceptor fluorescent dye. The internal conversion is detailed as a dotted line, while Dexter type transfers are shown as dashed lines. Dexter transfers-to elimitate-are schematised as dashed lines crossed out. After [bal 00].

290 Optoelectronics of molecules and polymers fluorescing dye singlet is formed, its radiative fall is immediate. This helps to reduce the total number of excitons in the system at anyone moment and, by consequence, the risk that they would transfer towards a triplet state and extinction . A widely used material for the host electron donor layer is a dicarbazolebiphenyl called CBP. Typically, the phosphorescent intermediate is iridium tris(2phenylpyridine) otherwise denoted Ir(ppy)3, and the fluorescent dye is often DCM2, a red dye. Complete chemical formulae can be found elsewhere [bal 00]. With an optimised structure, which contains a considerable number of transport layers of both holes and electrons, external quantum yields greater than 3% have been found, a multiplication by four of that obtained from the fluorescent dye alone. Another advantage of this configuration is that it does not rely upon phosphorescent emissions, which permits the application of molecules which are not necessarily phosphorescent as the phosphorescent intermediate [sam 00]. We can also note that the increase in radiative yield decreases the number of phonon descent processes, which in tum diminishes losses due to heating and, accordingly, increases LED lifetimes. There is however a cost, and that is that as the energy transfer descends towards lower and lower levels, and the resulting emissions are mostly in the red, sometimes in the green and with extreme difficulty in the blue. This tends to limit the number of available applications for these systems . In the fabrication of electrically pumped lasers we can see that this technique can be used to limit losses due to reabsorption of light emitted by triplet excitons . In addition, this by-passing of selection rules can be useful not only to optimise the number of singlet excitons produced but also in order to prepare organic lasing diodes.

4 Energy transfer with rare earths and infrared LEDs a Ligand-rare earth transfers at the heart of chelates The name chelate encompasses a group of compounds consisting of multi-dentate ligands attached to a central cation . Co-ordination bonds between the ligands and the cation are formed by the ligands donating electron pairs. As already detailed for the chelate Alq3, the standard chelate for electroluminescence, the emission colour is determined by the ligands with a peak at 525 nm. Doping with dyes can induce blue or red emissions. On inserting a hole transport layer at the anode side, luminances as high as 10000 cd m- 2 have been reached with yields of 1.5 lm W- 1 . Photoluminescence or electroluminescence can be obtained from chelated rare earths, attached to ligands by single or double bonds . Gadolinium yields UV light, while cerium gives rise to the colour blue. Terbium gives green, europium shows red, dysprosium displays yellow and erbium emits in the infrared . Figure X-28 schematises the basic mechanism. Either by photoluminescence or electroluminescence, respectively from incident photons or electron injection, ligand electrons are excited from a fundamental singlet to an excited singlet state from which relaxation results in fluorescence. Triplet states are also occupied due to excitation by electroluminescence . The excited states, whether they be singlet or triplet can go on to transfer energy to energy levels associated with the rare earth 4f orbitals, as long as Crosby's

X Electroluminescent organic diodes 291 L1GA, ... D

sing

nonra d ali ve

Figure X-28. Rare earth ion excitation by organic solids.

rule is respected [Cro 61]. This states that the lowest energy level of a ligand triplet state in the complex should be close to equal or just above the resonance energy level of the rare earth ion. A difference of 0.05 eV is sufficient to ensure a 95% photoluminescence yield . On emitting light or through non-radiative processes, the rare earth ion can then relax to its fundamental state . It is interesting to remember that when a ligand is excited from its fundamental state, 25% of the resulting states are singlets and 75% are triplets . All this energy can be transferred to the rare earth ion, which means that there is a potential internal quantum yield of 100% [Bel 99]. In these chelate systems, it is the rare earth which is responsible for their lifetimes and the wavelength of the emitted light. The ligand though is responsible for the absorption, the power of the emission and , in practical terms , the ease of handling. ELAM, a company in London, indicates that yields obtained with these systems can be up to 60 to 70lm W- I , which can be compared against that of fluorescent tubes at 100 1m W- I . OPSY, a company in Oxford, UK, has announced luminances of 200 cd m- 2 in the red and green, with photoluminescent yields of 80%.

b Infrared organic LEDs [Gil 99] Following purification, sublimation of a methanol solution of erbium chloride and 8-hydroxyquinoline can give films of erbium tris(8 -hydroxyquinoline) (Erq-), Er3+ exhibits a strong luminescence centred at 1.54 urn due to a transition from the first excited 4113/2 state to the fundamental 4115/2 level. Light at this frequency is within the window of transparency of silicon based optical fibres, used in communications. Using the OLED structure glass /ITO/TPD(50 nm) /Erq3(60 nm)/AI, photoluminescence around 1525 nm and electroluminescence around 1533 nm was obtained. Secondary emissions were also found and attributed to isomers of Erq3' There are certainly difficulties with respect to yields, and only qualitative results have been published (i .e. graphs with arbitrary scales) . The size of the difference in

292 Optoelectronics of molecules and polymers energy between radiative recombinations at the ligand s, which persist in Erq, and the 4113/2 ---+ 4115/ 2 line may explain these results.

5 Microcavities A typical structure is made using a Bragg mirror inserted between a glass substrate and a layer of ITO (98 nm and with n = 1.8), onto which a layer of PPY is deposited (160 nm). Aluminium or silver electrodes function as a semi-transparent mirror, as shown in Figure X-29-a. The photoluminescence spectra, like the electroluminescence spectra exhibit considerable narrowing of the spectral curve (having a width of only around 5 nm half way up) with respect to that of PPY obtained in free space [Tes 96]. The intensity of emissions permitted by the cavity are also increased greatly, and even more so when using silver as the mirror as it allows less losses than aluminium (Figure X-29-b) .

6 Electron pumping and the laser effect For the OLEO s so far detailed, the current densities are of the order of I to 10 rnA cm- 2 for luminances of around 100 cd m- 2 . For lasers, higher densities are required , but they are a priori limited by low carrier mobilitie s so that at high injection levels absorption increa ses. The use of tetracene (4 join ed phenyl rings) has allowed relatively high mobilities-at least for an organic solid!-at around 2 cm 2 y - 1 s- 1 to be attained. Amplified spontaneous emission by optical pumpin g was observed in the middle of the 70s [Pop 82] in anthracene and doped tetracene. More recently, a lasing effect by injection has been reported [Scho 00]. For the latter, tetracene was used in a perfect crystalline form in order to banish defaults and impurities otherwise insupport able in optoelectronics. Vapour phase deposition was used to give crystals of around I to 10 urn thickness and I mrrr' surface area. Two field effects at the electrodes were used to ensure an equilibrated injection of holes and

ITO DUR

(a)

Glass subs trate

Figure X·29. (a) Microcavity structure; and (b) general shape of photoluminescence curves.

X Electroluminescentorganic diodes

293

. :3

- VI Grid 1

1

<8> Figure X-30. (a) Cross sectional view of structure; and (b) schematisation of grid-source tensions applied to give field effect injections.

electron s (see also this point developed in Section l-b in this Chapter). Figure X-30 schematises the fabricated structure for this device and shows the two field effect electrodes used to inject charges. To describe the device we will refer to this Figure . Around the crystal there were deposited a source, a drain, a dielectric grid and a grid, with the former two for hole injections made from gold, while aluminium was used to inject electron s. Holes were injected by the application of a negative sourcegrid at the upside and electron s were injected at the lower side by the application of a positive tension between the source and grid deposited on the lower side. The field effect acts much like a variable doping at the contact s with the crystal, and the concentration of carriers can be varied by adjusting the grid tension. Between drain I and source 2, and source I and drain 2, a tension was applied (around 5 V) to a slice of the crystal allowing electrons and holes to traverse the device, respectively, upward s and downwards. Electroluminescence and amplified stimulated emissions are detected laterally. In addition, the organic crystal can play the role of waveguide, but the grid dielectric here is too thin at 150 nm to act as a coating . In fact it is air which plays this role, allowing only a multi-mod al guide. Cleavage of the crystal faces, which have a reflectivity of 8% can induce a Fabry-Perot type resonance. At a value Vg = 50V, carrier densities of 1013 cm- 3 were estimated using measurement s of capacity and variations in tensions and thus conductivities. A potential drop of 5 V held between the high and low points of the structure assure current flow. As a consequence of injection, excitons are formed and light emissions are observed. At a temperature of 5 K, three peaks were observed for a very low pulsed current « 1A cm - 2 ) . The peaks were characterised as being due to radiative transitions between first excited and fundamental states and included vibronic levels (0- 1, 0-2 .. . transitions). They were separated by 170 eV, a value corresponding to intramol ecular vibrations. As shown in Figure X-31, an increa se in current density, by increa sing the grid-source tension, gave rise to a dominant 0-1 band with a mid-height width reduced to lOmeV at a density of 30 A cm- 2 . At j > 500 A cm- 2 , a vertical line appeared, of width less than I meV. This abrupt narrowing of the spectral line at a threshold current of 30 A cm- 2 is typical of a spontaneously amplified emission with the contributed guidance gain. The second drop at j > 500 A cm- 2 corresponds to the start of the lasing effect.

294 Optoelectronics of molecules and polymers J~ Wid!h_o!~IU.i~ion

_100 (meV)

"

band at mid-height \

\ \

-10

-1 I

1

\

<,

-I

--,

\

\

\ \ \

\ 1'-

10 100 1000 current density (A em")

Figure X·31. Emission line width at mid-height and at 5 K with respect to the function of pulsedcurrentdensity (l0 I1S, 100Hz).

VII Applications in the field of displays: flexible screens 1 The advantages One of the potentially most interesting developments, in industrial terms, is the fabrication of flexible displays and screens . It was realised , right from the start of the development of screens with pixels made from OLEDs and PLEDs, that they had numerous advantages . (Recalling the term, pixel comes from the contraction of 'picture-element' , while interestingly in French the word 'eldim' would be used, a similar contraction of the words 'element d'image' .) When compared to liquid crystal displays (LCDs), they have the advantages of giving a wider angle of view (Lambert 's emission) , a higher luminance (several 100,000 cd m- 2 have been attained) , an extremely high commutation speed, a low minimum working voltage around 5 V, low levels of energy consumption and no need for back-lighting. There have been plans to use OLEDs as back-lights and white diodes have been prepared for the lighting market. We will first say a few words on white diodes , before looking more in depth at the realisation of screens.

2 The problem of ageing Ageing determines the lifetimes of LEDs [Ngu 98 and Miy 97]. The definition used to qualify the lifetime of a device is the time it takes for the luminance to descend to one half of its original value [Si 97]. The measurement can be made using either of two methods . In one the applied tension is kept constant, and in the other the current which traverses the device is retained constant. Figure X-32 shows the results obtained using the latter method . The original luminance is fixed at around 100 cd m- 2 . The tension and the luminance are measured every 30 seconds so that the curves L = f(t)

X Electroluminescent organic diodes Luminance (cd m-

z)

30 nm CUPC + 60 nm Alq3

I

295

Voltage (V)

14

120

12

100

-Luminance -Voltage

80

10 8

60 6

40

4

20

2 Time(h)

0

0 0

Figure X-32. L

3

= f(t)

6

and V

9

12

15

18

= f(t) characteristics of the ITO/CuPc/ Alq3/Ca/ Al structure.

and V = f(t) can be traced . The example structure shown, ITO/30nm CuPc /60nm Alq3/Ca, has a half-life thus estimated as 12 hours . Outside of problems due to breakdown, which can result from an inhomogeneous film, migrations from the electrodes towards the layers can occur, stimulated by field effects. As examples we can cite metallic indium which comes from ITO, as does oxygen which can go on to modify chemical groups , most notably those which fluoresce. At the cathode we can blame oxygen and water vapour for the appearance of black points localised about defaults which were there to start with and just go on growing . Another participating fault is no doubt that of crystallisation of active layers due to heating . This is the reason why there have been numerous works published concerning the treatment and improvement of electrodes and interfaces, most notably those of ITO. An example of one treatment is that of subjecting an ITO surface to oxygen plasma for 10 minutes which results in an improvement in its workfunction and this an improvement in hole injection. The roughness and the resistance of the ITO surface are also improved using this method and the overall yields are improved [Kim 99]: Finally it is worth mentioning the unavoidably necessary process of encapsulation of organic LEOs. Different approaches have been used: that of sealing between two glass plates and a ribbon of glue; and encapsulating within a polymer barrier layer such as PPX or one of its chlorinated derivatives deposited using VDP (see Chapter VIII for further details) . In common with all electrical components, organic diodes need to be fabricated a priori in a dust free ' white room ' so that the lifetime required for industrial applications is attained, which at present is of the order of 10,000 hours . Indeed, on an industrial scale, the fabrication of specific layered devices requires the connection of individual chambers each operating specifically to a certain layer. At the present moment, each company will have without doubt its own proper techniques.

296

Optoelectronics of molecules and polymers

3 The specific case of white diodes Uniax, within the EEC, has fabricated PLEDs for lighting purposes which have photon to electron yields between 2 and 3% operating at 3 to 4lm W- I at a low applied voltage . The luminance is better than that obtained from incandescent or fluorescent lamps . As ever though, the problem remains lifetimes, mostly due to heat. In a pulsed mode, luminances close to 100,000 cd m- z have been reached. If initial yields of25% can be attained, not an entirely unrealistic value, then this area awaits considerable development. Figure X-33 schematises the 'intelligent' layering of emissive layers with each emitting in the visible region . The ordering of the layers is made with respect to the type of injected carriers and the direction of emission, towards the emission window, so that successive absorptions do not occur. In effect, the layer nearest to the window should have the largest band gap [101 97 and Miy 97].

4 The structure of organic screens One of the advantages of organic LEDs is their very short communication time . Inversely, that also means that their remanence is very short lived. So, in the case of multiplexing, where sweeping of each line of pixels is required (as in passive matrices) and so the screen gives an acceptable average luminance (from around 100 to 200 cd m- z for a television), the luminance at anyone moment needs to be very high (n times the number of lines swept, for a screen with n lines) . The lifetimes of LEDs functioning under such conditions is therefore well reduced. The use of active matrices therefore becomes inevitable. They also offer a gain in resolution. This change in direction is also justified by the fact that OLEDs are particularly well suited to individual addressing, which can be performed via a polycrystalline silicon FET, although organic semiconductors can claim also to be perfect candidates as materials for the commanding transistors [bur 91 and hor 00]. In Figures X-34-a and -b, respectively, are shown a schematisation of a MIS field effect transistor (MISFET), which has a semiconducting channel made from PPY implanted with iodine ions (E = 30 keV, D = 4 x 10 10 ions cm- z) and its characteristics. This transistor operates in the accumulation phase, with the gate voltage being negative with respect to the p-type semiconductor finally obtained [Pic 95] .

~

Cathode Al

.... ... ~

,r

Blue

(yrcc1

Red

Glass

.

Multi-laver Iq3 (40 nun ) Alq3: i1eRed 1% mor 1(5 om) Alq3 (5 om ) TAZ (3 om ) TPD (40 om ) TO

Figure X-33. Schematisation of a white LED with the required order of layers.

X Electrolum inescent organic diodes

PPV

297

(a)

gam

16 g t: <3 c

4

1---------.__

~ 2

~

~

0 ..............·-

-8

-

_

.·

-6 -4 ·2 SourcclDrain Volt age (V)

cathode

Alq TPD

ITO

Figure X-34. (a) Configuration used in a polymer based MISFET transistor (after [pic 95]); (b) Ids = f(V ds) characteristics following Vgs for a MISFET with an iodine implanted PPV semiconductor channel; and (c) schematisation of a completely organic pixel.

Figure X-34-c shows the general constitution of a completely organic pixel from a monolithic integrated polymer FET and an organic LED [Bao 99]. The ITO, placed against the substrate, makes the LED anode while a photographically imposed layer of gold defines the gate electrode of the FET. The insulator for the gate is a layer of Si3N4 which is about 180 nm thick . The semiconductor, poly(3-hex ylthiophene) (P3HT) is a regio-regular polymer helping give it a high charge mobility at around 0.002 cm 2 V -I s -I. Above this is the drain and the source and the channel is about

298

Optoelectronics of molecules and polymers

5 urn long and 1 mm wide . The drain is joined to the anode of the LED, which is of a classical bilayer structure and has en external yield of about 0.4%. The maximum current through the LED is 50 itA cm- 2 (equivalent to a current density of 72 rnA cm- 2 ) . The luminance of such a pixel has been estimated at 2300 cd m- 2 , a value well above the 100 cd m- 2 necessary for display applications. An all polymer pixel has also been described, in which PPV replaces Alq3 and polyimide replaces SbN4 as the dielectric grid. As for the deposition techniques used, we can say that small molecules are deposited using evaporation under vacuum and polymers are spread using spincoating. The technique of ink-jet deposition which has appeared is particularly adapted to polymers, while there are also techniques based on 'roll-on-roll' for small molecules [Bur 97]. It is interesting to consider that while colour screens require the three colours red, green and blue (RGB), the necessary materials have been identified (although need some improvement [den 00)), and that all is required is juxtaposition of the three types of pixels using techniques particular to organic solids, which are not that removed from those currently used in microelectronics [Bar OOa and Bar OOb].

5 A description of the fabrication processes used for organic RGB pixels Research into improving OLEDs lifetimes to greater than 10,000 hours with initial luminances of more than 300 cd m- 2 also stimulated a development in the technology surrounding their use, specifically, in flat screens . To fabricate the RGB pixels, various technologies were tested, particularly to try and avoid etching based techniques: • • •

development of white diodes which emitted through RGB filters; use of blue diodes (i.e. at the highest energy required) which excited phosphors emitting in the green and red; and adjustments, using micro-cavities, of emissions from a wide band organic emitter.

However, for the above systems, the power emitted was limited, and the use of three individual layers for each colour was found to be indispensable. This required the fabrication of sub-pixels using etching techniques: • •

' wet' technology using solvents, either acids or water, which nevertheless generate problems associated with the induced degradation of the active organic materials; 'dry 'technology in which pixels are obtained using plasma etching, which a priori, presents no specific problems due to degradation.

In order to obtain high resolution screens, anodes and cathodes necessarily should be made to a precise geometric form, and this in accordance with the structure of the pixels on the screen which are either deposited in a matrix of points or as segments . The ITO anode is the most facile to prepare, using either 'dry' or 'wet' etching techniques being done before any organic layer is deposited. For the cathodes though, which demand an extremely high degree of geometric precision, their fabrication requires extreme care . In reality, the materials used for the cathodes (AI-Li or Mg-Ag

X Electroluminescent organic diodes

299

for examples), can be deposited directly onto an organic layer using 'wet' etching, however, when the organic layer is one which is fragile and sensitive to thermal or solvent induced effects , then particular steps must be taken. The literature proposes various routes, depending on whether small molecules or polymer s are involved, and below we shall look at some of the possible methods .

a For layers from sublimed molecular materials: use of aT-form cathode separator [Nag 97] The technology should allow for the fabrication of OLED devices which have dimensions around I urn . The technique involving T-form cathode separators was used by Pioneer to realise their first commercially available screens . The separator between pixels, sized around 30 urn , was prepared prior to depositing the organic layers and the metallic cathode. Practically speaking , the cathode is fabricated around the T-form separators in three steps, as shown in Figure X-35. The cathode separator is shaped in conical form and has a retracted base-as shown in Figure X-35-a . It can be obtained using a negative working photo-polymer-the irradiated part of which resists etching development-by relying on the cumulated exposure with depth . The photo-polymer, deposited by spin-coating, exhibits properties similar to those of a photo-resin

substrate (a)

substrate (b)

(c)

Figure X-35. The three stages in diode fabrication via a T-form cathode separator: (a) elaboration of the cathode separator; (b) evaporation of organic materials at an oblique angle with respect to substrate ; and (c) evaporation of the metal cathode at a angle perpendicular to the surface.

300

Optoelectronics of molecules and polymers

Photo- oolvrner

...,..... 1'h00o·oolvmer

Substrate (a)

Substrate (b)

Slow development 4 "" Fnsl development

Figure X-36.Realisation of the phasegivenin FigureX-35a: (a) spreading thephoto-polymer; (b) selective exposure throughmark; and (c) development.

with additional properties in development speeds and exposure time. As shown in Figure X-36, a longer exposure time can result in a more noticeable removal of material. The T-form of the separator can also be obtained using a double polyimide/Siolayer deposited on the anode (polyimide on the anode then SiOz on the polyimide). The SiOz layer is etched using a dry technique (to yield a anisotropic form with vertical walls) of selective exposure through a mask (photolithography). The polyimide , which is not sensitive to light, is then etched using a 'wet' process (thus isotropically) as a sub-etching, thus giving rise to the vertical 'leg' of the T. Organic layers are then evaporated, obliquely, onto the surface, as shown in Figure X-35-b , and surround the foot of the cathode separator. The metal layers are then finally evaporated perpendicularly to the substrate, schematised in Figure X-35-c, in such a way as to never touch the bases of the separators. Adjacent cathodes are thus electrically isolated from one another. While the T-form separators were prepared with dimensions of 30 11m x 30 11m [Nag 97], with present day photolithographic proce sses, dimensions as low as 10 11m x 10 11m may be reached. A passive matrix screen was prepared 256 x 64 points with pixels sized around 340 11m x 340 11m. The geometrical placement of cathodes can be rectilinear or curved, another advantage with respect to various fabrication methods.

b Integrating three RGB polymer LEDs The technology used for fabricating screens must be able to integrate sub-pixels. Polymer based orange, green and blue pixels have been prepared by sequential sampling of three polymer films on the same glass substrate covered with ITO, over which an insulating layer had been placed. The polymer films were deposited using spin-coating and then covered by a metal electrode, itself deposited by evaporation under vacuum . The last layer served as a self-alignment mask for when the films were subjected to plasma etching to expose adjacent polymer sub-layers [Wu 96]. Thus it is not the organic layer which defines the dimensions of each pixel or sub-pixel (or the LED), but rather it is the electrode (here the cathode metal or, otherwi se the ITO anode). The structure is schematised in Figure X-37 . It should be noted though that the unprotected sides of the polymer layers deposited first can be attacked by chemical agents,

X Electroluminescent organic diodes

301

Polymer

R

G

B

Figure X-37. Set-up of three sub-pixels with cathode defined.

for example the solvent of the seco nd polymer layer, which may be required for a different colour. Other than possibly forming geometrical deform ations, this process can also result in degradations and electrical short-circuits. In order to limit and even compl etely stop these problems, the structure can be modified so as to protect the side of the polymer layer. Figure X-38 shows that in order to do this, an insulating layer is depo sited onto the ITO layer, and a window which allows acce ss to the ITO layer is opened at the insulating layer, which defines the active zone. In practical terms, the insulating film is made of a nitride of silicon deposited by PECVD at 250 °C and has a thickne ss of around 100 nm. The active window is obtained in the SiN x using standard photolithographi c etching processes. For the first set of sub-pixels, the polymer film (ora nge following the particul ar reported case) is spread over the whole surface of the scree n and the cathode metal (AI included) for this layer of sub-pixels is deposited by evapora tion under vacuum (:::::: 10- 6 torr) through a mask. The structure is then exposed to oxyge n plasma which etches away resins and isolated organic materia l but leaves intact zones below the AI. This plasma etchin g is in some senses self-aligni ng and does not require a supplementary masking stage . In addition, its use negates the risks of overexposure of the organic films to solvents that would otherwi se be encountered in a 'w et' process. An additional layer of Al can be deposited throu gh a mask to seal the device, as shown in Figure X-39. Thi s takes the sides of the organic layers out of harms way from organic solvents, air and hum idity. We can also note that this techn ology can be extended to small molecules. The following green polym er layer can then be deposited to gain the second pixel. The above detailed stages, up to the sealing layer, are then repeated.

AI Polymer lnsul ling film_ (Si

',J

. -_

_

_

-tot--

l1 ~ ~

R

v

B

Figure X-38. RGB structures prepared to protect pixel edges.

Cathode metal

ITO

302

Optoelectronic s of molecules and polymers

Pnlymcr In ul tin~ film lSi, ',)

~~==--

I Carbooemct.Ill ITO

Figure X-39. Sealing sub-pixels with a layer of aluminium.

With mechanically aligned films, typicall y, dimen sions for the active zone are 2 x 2 mm 2 and the metallic contacts are 4 x 4 mm". The zones of the metallic contact go beyond that of the active material giving some space between the side of the system and the electronically useful zone. This configuration has been tested with success under a dry nitrogen atmosphere , without loss of characteristics . This technique may be extended to pixels of smaller sizes (hundred of microns ), to fabricate high-resolution screens, using classic photolithographic technol ogies. Excepting the transfer of samples from the etching chamber, all preparation steps should a priori be effected within a glove box under a dry nitrogen atmosphere. The technology described above has been used with success, and without altering optoelectronic characteristics, to prepare an integrated structure with PVK as the hole transporter, doped PVK as the emitting layers, layers of Alq3 or PBD to transport electrons and the following various dyes: PVK/Alq3/nile red for red (orange) sub-pixels; PVK /Alq3 /coumarin-6 for green sub-pixels; and PVK/PBD/coumarin-47 for the blue sub-pixels.

c Ink-jet technology [Bha 98] The techn ique of polymer spin-coating presents, in fact, a considerably number of disadvantage s with respect to preparin g pixelated screens. Normally, nearly 99% of the polymer solution is wasted and it is difficult, if not impossible, to prepare lateral faces, Deposition using ink-jets could therefore appear to be an excellent substitute method, with only less than 2% of solution lost. This technolog y also allows treatment of large surfaces, is compatible with the indispensable addressing of (coloured) pixels and is relatively cheap to operat e. Certain precautions are necessary, however, to ensure a regular deposition of polymer solution. The use of dilute polymer solutions was evaluated , with deceptive results with respect to the quality of the components. Eventually a hybrid technol ogy was adopted which uses an absorbent , intermediate layer, which can be made of a host polymer, and has a thickne ss of around 100 nm. The intermediate layer exhibit s three principal characteristics: • • •

it smoothes out bumps and dips which can be generated by the ink-jet ; it acts as a sponge, virtually fixing the polymer solution; and if it is formed of a semiconductor with a large band gap, then a multi-colour emission can be obtained by energy transfer when the ink-jet deposited material has a smaller band gap.

X Electroluminescent organicdiodes

303

In order to fabricate colour screens, red, green and blue polymers are used and the intermediate layer can be an electron injecting layer. Inversely, good results can be obtained by using the ink-jet process for the charge injecting layer and with a luminescent polymer for the intermediate layer. The emitting zone is thus defined by the charge injection layer deposited directly by the ink-jet printer. Figure X-40 shows a cross-sectional scheme of a system with the hole injection layer deposited using an ink-jet printer. A commercial printer was used with an aqueous solution ofPEDOT [bha 98]. The active polymer, MEH-PPV, was deposited using spin-coating and serves also as the absorbing, intermediate layer (120 nm thick obtained from a 1% solution of MEH-PPV spun at 2500 turns per minute). At 5 V, the 'logo sandwich' ITO/PEDOTIMEH-PPV ICa gave rise to a luminance of 200 cd m- 2 with a contrast close to 800 (the luminance ratio of emitting an dark zones) . Given the ease of controlling the ink-jet printer and its depositions, a practically infinite range of logos can be made and writing, in particular, can be computer generated. In addition to variations in colour, different grey levels can be attained by changing the superficial concentration or size of the deposited spots . Typically, plots of sizes between 180 and 400 IJ.m were possible in the original experiments. For monochrome screens, dimensions of the order of 100 IJ.m suffice. Ink-jet technology is thus well placed to be used in the fabrication of screens and this is without having to define pixel positions by overlapping lined anodes and cathodes in columns, for example. As another example, we can cite the bicolour screen [Cha 98] which was fabricated by directly spin-coating an intermediate layer onto ITO, and then ink-jet printing was used to deposit a polymer that had a lower band gap and was capable of diffusing through the intermediate layer. The intermediate layer was made from a polymer with a large band gap, was soluble in water and emitted in the blue part of the visible spectrum. The material that was ink-jet deposited was also water soluble but emitted red-orange light. On absorption, there was a transfer of energy from the former to the latter, and red-orange photoluminescence or electrolumincscence could be obtained from the ink-jet pattern, as detailed in Figure X-41. A tricolour prototype has been presented by CDT-Seiko-Epson, as shown in Figure X-42 . Its structure was based around series of pillars, formed from strips of polyimide (Figure X-42-a) deposited onto ITO and placed so as to leave spaces to receive ink-jet deposited drops of solution. After adapting the wetability of the

Cathode (Ca)

.-

.-

~ I ectroluminescent

polymer (MEH-PPV)

-Conducting polymer (PEDOT) to.node

f---

(ITO)

Substrate

Emis sion

Figure X-40. Pixel with an emittingzone defined by the charge injection layer (here, PEDOT depositedby ink-jet process.

304

Optoelectronics of molecules and polymers athode Orange polymer deposited using inkjet Diffusion of orange polymer within the layer Anode

I

EI. blue

EL urunue-red

Figure X-41. Hi-coloured pixelsobtained following excitation of a polymerby anotherwitha largergap.

polyimide by 02 and then CF4 plasma treatments, into each pixel caisson, constituted from three sub-pixels, were placed in the following order: • • • •

PEDOT by spin-coating into the base of the caisson; a red emitter and a green emitter by ink-jet printing into respective sub-pixels, as shown in Figure X-42-c; a blue emitting polymer by spin coating, as in Figure X-42-d; and a cathode (Ca then AI) as in Figure X-42-e .

6 Emerging organic-based technologies: flexible electronic 'pages' As often occurs in the course of scientific development, the electronic-ink (e-ink) technologies were developed in parallel to that of the electroluminescent (EL) organic screens. Given that most people prefer reading ink deposited on paper, it is not unimag inable to think that 'e-ink' systems will replace EL screens, at least with respect to certain needs . A group at the University of Groningen proposed the use of e-ink systems in the preparation of flexible screens , which might be used as electronic pages [ber Ola] . We could call these 'e-pages' . This technology could be adapted to the fabrication of screens operating either by transmission or by reflection, and uses as base a thick plastic film covered with micro-holes each 10 to 20 urn in diameter. The holes resemble pixels once they are filled either with a light emitter or by an e1ectrophorescing ink. For the latter, a coloured solution which contains white particulates is used. After sealing the pixels , applied electrodes can control the movement of these particles, directing them either towards the surface or the base of the plastic film and thus changing their appearance from white to coloured. The German ' start-up' company, Papyron, which developed this technology is in competition though with an American technology, developed notably by E Ink Corporation) which is based on capsules. The capsules, again between electrodes, are about 100!-tm in diameter and are filled with a dye and a white , negatively charged chip. On applying a positive tension to the upper surface electrode, a capsule appears white as the chip is attracted towards

X Electroluminescent organic diodes

305

ITO transparent anode

... .. . .. . .. . .. . .. ...

Strips of polyimidc

(a)

(b)

Red (rhodamine IOIIPPV) and green (PPV) cmiuing polymers deposited by Ink-jet

(c)

Blue emitting polymer (polydialkylfluorenc) deposited by spin-coating

(d)

Cathode deposited by evaporation under vacuum Ca then AI

(e)

Figure X-42. Steps in the real isation of a tri-colour prototype developed by CDT-Seiko-Epson.

306 Optoelectronics of molecules and polymers the screen. According to agreements made between E Ink Corporation and Philips (from the journal Opto Laser Europe, vol 5, page 84, 2001) the ink can activated using electronic controls delivered via an active matrix developed by Philips. Papyron insists that its technology exhibits numerous advantages over that of E Ink. Papyron states that its system allows a wider angle of view due to its pixels being flat rather than curving inwards and that its system allows a simpler fabrication of colour screens by filling different holes with different colours, an objective which can be realised using ink-jet technology. Thus, each pixel can be individually and electronically controlled using plastic transistors or other variants such as poly-crystalline silicon based transistors. For more stable displays, a liquid which exhibits wide variations in viscosity with changes in temperature can be used to facilitate image displays, as once the image is formed at a temperature high enough to allow easy movement of coloured particles , then the image can be 'fixed' by decreasing the temperature of the pixels, an operation which can be controlled by laser, for example . For the present time, most companies concentrate their efforts on developing systems which can be fabricated using 'roll-on-roll' techniques . A research group in Limoges, France, is currently studying the formation of micro-holes through ionic columns due to field effects, in a system probably well adapted to automatic control of the depth of micro-holes for pixel fabrication. Xerox at Pali Alto have invented a technology called Gyricon which may compete in the same domain [ber Olb]. This procedure uses a thin plastic film into which is randomly inserted small pearls. The opposite hemispheres of these pearls are oppositely charged and are black on one side and white on the other. Depending on the charge applied , the spheres present a black or white surface. A continuous application of charge is not required; the spheres do not rotate back to a certain position; their power consumption is thus relatively low. Given that this sort of technology is not sensitive to air or humidity, it can be easily transferred to large surfaces. However, the time required for each sphere to change shade is relatively long and the integration of colour (using the three primary colours) in the system is hard to envisage, so this technique may not be useful for video applications. Nevertheless, such a system is sufficiently effective to enable rechargeable electronic books, a possible market worth over US$1O billion . Other possible techniques have been evaluated, notably by Lucent which developed a polymer based 25" screen using transistors fabricated through embossing processes.

VIII The prospective and actual production at 2002 Most of the following information has come from issues of Opto Laser Europe at the end of 2001 and details the actual commercial and industrial situations at this time. The market for flat screens is worth several billions of US dollars, and concerns both e-ink technologies, which necessitate external light sources, and emitting screens based in particular on organic electroluminescence, which do not. The market estimated for the year 2000 was US$24 million, a 100% increase on 1999, while the

X Electroluminescent organic diodes 307 year 2005 is expected to encompass a market worth US$3 .3 billion . An explosion in this market is predicted for the period 2006 to 2010 . The main aims for those economically involved in this market concern the fabrication of screens which are lightweight, flexible and exhibit high luminance, contrast, wide angle of view, have low energy consumptions, and last but not least, cost little to make . The target market includes applications such as portable telephones, digital cameras, televisions - which can be hung up on walls or rolled up, tables which can act as displays, programmable and therefore rechargeable books and even wallpaper! Pioneer, with a licence from Eastman Kodak, commercialised, in September 1997, the first organic screen , which was destined for an auto-radio facade. It was based on Alq3 and had dimensions of 9.5 x 2.1 em, and consumed 0.5 W. LG-LCD in Korea announced the fabrication of a VGA screen functioning at 5 Y and displaying a contrast of 100:1. The screen was green and had a luminance of at least 200 cd m- 2 . Uniax fabricated small screens for portable telephones which emitted orange light at 610 nm, operated at 10V and required only I rnA per pixel to yield a luminance of 200 cd m- 2 . This screen was based on small molecules, much like the first colour screen introduced by Pioneer for a portable Motorola telephone. In May 2000, Sanyo Electric associated with Kodak presented an active matrix screen of 5.5" consisting of 240 x 320 pixels. The American company Universal Display Corporation (UDC) , risen from Princeton and South California universities , subsequently also developed screen s based on small molecules. On turning to technology based on polymers, a highly involved company is Cambridge Display Technologies (COT), which grew from the Cavendish Laboratory at the University of Cambridge, in which macromolecular electroluminescence was discovered. The firm Covion has announced an annual production of 40,000 litres of conjugated polymers, destined to be used by COT and by Philips, the latter having a factory in Heerlen (The Netherlands) producing screens based on light emitting polymers (LEP)s for portable telephones. In March 2000, COT and Seiko announced the realisation of a R-G-B screen based on polymers deposited using ink-jet methods . The fabrication, however, posed more problems than initially supposed, with incompatibilities arising between the polymer inks and the ink-jet deposition procedure. Collaboration with Seiko Epson brought about the presentation of a colour prototype in May 2000 and then again in May 2001 at San Jose as part of the Society for Information Display 's conference there . Lifetimes greater than 50,000 hours have been attained using adapted encapsulation procedures. COT has agreements with the Japanese firm Tokki for vapour phase physical deposition procedures and encapsulation, according OLE in July and August 200 I. A pilot line worth US$25 million was built in Godmanchester with the aim of fabricating 5 million screen units per year. The effective production was planned for the start of 2002 with 'roll-on-roll'procedures taking the place of ink-jet methods . At the University of Arizona, the ink-jet method was applied to screen fabrication and yields of 0.9% were obtained. Sony though announced the fabrication of a 13" diagonal screen with 800 x 600 pixels (SYGA), R-G-B colour and a peak luminance of 300 cd m- 2 , the device controlled by an active matrix formed from polycrystalline silicon.

!

308 Optoelectronics of molecules and polymers CDT's polymer technology has also been developed by Hewlett-Packard and Philips. The latter company has, in Heerlen, developed a pilot line for PLEDs on a glass substrate, as the flexible films were found to not entirely fulfil the role of barrier against humidity and oxygen. Philips consider that in addition to their higher luminance and wider viewing angle, organic LEDs are preferable to liquid crystal displays because of their shorter response times. By mid-2001 it was planned to realise a monochrome display for portable phones of size 65 x 100 mm and capable of displaying video images . Nevertheless, DuPont in the USA has announced a plan to develop and fabricate plastic substrates for polymer based screens on active matrixes with Philips and Alien Technology, another British 'start-up' company. DuPont have also planned, in co-ordination with other new companies, to develop more classical technologies using active matrixes on glass in China and the Philippines. The German company Osram, bought up licences from CDT and Uniax (a company started by A. Heeger and subsequently bought by DuPont) and has installed a PLED pilot line in San Jose (USA) and planned to start production in Malaysia during 2002. Japanese firms are not sitting back. Sony announced the prototype of the largest OLED screen, Toshiba planned to be selling in the market place polymer based screens by April 2002 and Tohoku-Pioneer and Sharp began an association to invest 424 million Euros into the production of organic EL screens . NEC and the Korean company Samsung, meanwhile, also formed an alliance to invest over a 5 year period 472 million Euros for the production of similar screens in South Korea from the Autumn of 200 I onwards. In England, outside of CDT, there is another relatively young company called Opsys based in Oxford which seems to be going through considerable expansion, with at the time of writing this book, 25 employees and 15 University researchers financed. The company first worked on organometallic luminescent material s, including transition metals and rare earths, and then on macromolecular materials such as dendrimers. At the same time, the company expanded in different areas, notably being the first in Europe to take up licences to exploit Kodak's small molecules . Opsys has also been developing new small molecules with greater stabilities and yields than those of Kodak, and, according to the Director, the absence of the necessary competencies within Europe have meant that they have had to go to Fremont (USA) to produce their screens. From 2002 onwards Opsys planned to fabricate their 1/1 to 5/1 diagonal screens , with active matrixes and back-lighting for portable devices. Larger screens , with appropriate licensing have been planned . The technology of the present time-based on the evaporation of small molecules-allows for little flexibility (for example with respect to changing substrate size), and it is planned that it will be changed at some point to a technology such as ink-jet or spin-coating based on polymers. The company does not see the choice between small or macro-molecules being one of chem istry or even optoelectronics, but more to do with problems associated with fabrication and costs. In addition , it seems that Opsys has already at its disposition a new class of polymers which do not fall under the umbrella of eDT's patents .

X Electroluminescent organicdiodes

309

IX Conclusion The industrial competition is wide open and the diversity of competitors currently operating is without doubt justified by the existence of such an enormous, and financially consequent, market. The market goes beyond that reached by liquid crystals and is planned to be used not only in conventional systems such as screens but also in a wide range of other devices, for example aeroplanes, cars, display panels, electronic books and in teaching systems. While the field of organic electroluminescent diodes remains young (for the most part being developed in the last 10 years), it is already vast and the number of annual publications on the subject is considerable. We can remark that a decade of intense research has actually given rise to a new source of light which is now ready to enter our everyday lives. Organic solids, largely studied during the 20th century with relatively few actual practical applications have now found in the form of electroluminescent diodes an enormous field of applications, a field which should continue to grow. In this Chapter, we have tried to present the essence of current knowledge. However, it would be highly pretentious to imagine that we have covered everything. As a useful indicator of the results of ongoing research, it is worth mentioning the reference [tro 01] which was prepared in collaboration with Thales and concerns the preparation of an OLED emitting blue-green light. A four layer structure, shown in Figure X-43 , combines a hole injection layer (HIL) made of PEDOT, an emitting hole transport layer (TNATA), a hole blocking layer (HBL) made of BCP and an electron transport layer (ETL) made of Bebq2, which has been shown to be more efficient than Alq3. These molecules and their chemical formulae are further detailed in Appendix A-II . The structure ITO /PEDOT/ TNATA/BCP/ Bebq2/ Ca, emitting in the blue-green at 508 nm gives rise to a luminance of the order of 12980 cd m- 2 operating at 11.2 V (Figure X-44). In the green though a greater luminance, more than 20,000 cd m- 2, was obtained using a tri-Iayer structure detailed in Figure X-22-b. To conclude, these results show that luminances obtainable with OLEDs can be particularly high and can satisfy the demands of a wide range of applications.

~

Cathode

v

Electron tran sport layer (ET L) Hole block ing layer ( HBL) ~---

rlt==========t: : : :;--

1 - - ' ' ' ' - - - - - - - - - - - - - ;_ _

Emitting laye r(TPD. ' PB. T ATA ) Hole inj ectio n layer (I ilL ) Anode (ITO) Glass substra te:

Figure X·43. OLEDobtained from a four layer structure.

310

Optoelectronics of molecules and polymers 12 000 Luminance (cd m· ~T

10 000

8000

2 )

'AT A

--6-

PEDOT+T . 'AT A

-

PEDOT+T. 'AT A+RCP+Alq3

-+- PEDOT+T. 'AT A+BCI' +Bcbq2

6000

Current density mAcm'!)

SOD

600

700

800

Figure X-44. L = f'(J) curves for various structure based on TNATA, including, notably, the structure ITO/PEDOT/ TNATA/ BCP/ Bebq2/ Ca.

As an example is shown the prototype screen for a portable telephone fabricated by the Briti sh firm Opsys using small organic mole cular technology, a technology in which Kodak Eastman hold s numerous patents. The screen has diagonal size of 1.24", is 1.8mm think and exhibits a luminance of 40 cd m- 2. It consists of 128 x 64 pixel s (image from Displays Europe, February 2002).

X Actual state-of-the-art and prospectives During the Society for Information Display (SID) meeting in Baltimore , USA , in May

2003, two firms claimed to have built the large st OLEOs yet. IDTech demonstrated a 50 em diagonal displa y driven by "super amorphous techn ology", a technology with a cos t much lower than this incurred with polycrystalline tran sistors. It had previously been thou ght that amorphous silico n was incapable of deli verin g sufficient current to

X Electroluminescent organic diodes

311

Figure X-45. (a): Kodak camera EasyShare LS 633; and (b) COT concept for a TV watch usingits light-emitting technology.

drive OLEO pixels, but IDTech claims that its display consumes half the power of a typical high-end LCO , and offers better colour saturation and a wider viewing angle . It has a resolution of 1280 x 768 pixels and draws 25 W of power at 300 cd m- 2 . Sony 'exhibited a substantially bigger active-matrix OLEO display at 60 em. It was comprised of four seamlessly joined pieces (displays) using a new tiling technology. The Sony display, at 1024 x 768 pixels, had a slightly lower resolution than the IOTech screen . Both companies say that their developments could lead to OLEO displays being used in televisions. Furthermore, Kodak and its manufacturing partner Sanyo showed a 15 inch OLEO screen which will be commercialised in three to four years [Ole 03-b]. The first commercialised large scale OLEO display (2.2 inch diagonal) was inserted into the Kodak Easyshare LS633 camera (Figure X-45-a) sold with a price tag of around €450. The display, with a viewing angle of 1650 has a resolution of 512 x 218 pixels and a power consumption of around 300 mW. It was developed by SK display, a joint venture between Sanyo Electric and Kodak. SK will initially produce around 100,000 two-inch OLEO displays each month , and hope to boost this value to I million at the end of 2003 [Ole 03-a1. Meanwhile, Phillips Electronics hopes to sell more than 100,000 electric shavers incorporating a small monochrome

312

Optoelectronics of molecules and polymers

OLED display in 2003. Phillips have already exhibited a 1.4 inch passive-matrix monochrome PLED at SID 2002, held in May 2002 at Boston, USA. Cambridge Display Technology (CDT) of the UK and the Japanese firm SeikoEpson developed in a joint company, Polyink, an inkjet printing technology based on light emitting polymers . They have fabricated a TV watch, shown in Figure X-45-b, and commercialisation is in progress. However, an opportunity for CDT to buy up Litrex, a Californian company which also manufactures industrial inkjet equipment, may lead to new directions being taken. It should be mentioned that displays made by Pioneer for car stereos and cell phones have been on the market for sometime already. Research into OLEDs has given-apologies for the pun-a vision of what may be possible in the future. Flexible displays could one day depict an electronic newspaper or, as curved screens , furnish the automobile or plane cockpit of tomorrow. 'Electronic ink' is an alternative way to produce soft electron ic newspaper and the E ink company of Cambridge, Massachusetts, USA, has demonstrated the world's thinnest activematrix display (0.3 mm thick) . The construction combines a thin, shatterproof steel foil transistor substrate with E ink's paper-like electronic-ink display material coated onto a plastic sheet. In 2003 the OLED display market is valued at around $500 million, and according to various market analysts (DisplaySearch, Stanford Resources, iSuppli) , may reach $3 billon (€2.7 bn) in 2007 or 2009 [OLE 03-a or Sie 03].

XI Organic photovoltaic devices

I Principles and history of organic based photovoltaics 1 General points: the photovoltaic effect a The photoelectric process In general terms, photoelectronic processes correspond to those engaged in the absorption or emission of light by a material. Solids can be subject to various effects , such as photo-conduction (conduction once the material is illuminated) or the photovoltaic effect (generation of a tension by illumination), and these effects are directly tied to transport mechanisms. Here we shall concern ourselves expressively with the second process . Within a photovoltaic system , we can see that following exposure to a ray of light, charge carriers can be generated by: • • •

electron band to band transitions (valence band to conduction band); electron transitions from localised states in the forbidden band up to the conduction band, or from the valence band to empty localised states in the band gap ; and the generation of excitons following their dissociation or their ionisation.

The first process can be found in inorganic crystalline solids, the second in inorganic amorphous solids, and the third, most commonly, in organic solids. Whatever the process though, the photovoltaic process consists of 3 steps which are: • • •

photogeneration of charges, by one of the three mechanisms above ; the separation of those charges and their transport (and eventual multiplication) to the physical limits (terminals) of the device; and the realisation of an electrically active contact that can give rise to a signal which can be analysed either as a phototension with the circuit open , or as a photo-current with the circuit closed .

314

Optoelectronics of molecules and polymers

b Various configurations used for collecting photogenerated charges For a photovoltaic devi ce to operate , the second step mentioned just abo ve is essential and can be cau sed by several different effects [Kao 8 1]:

a The volume or mass photovoltaic effect Thi s effect can arise from the diffusion of charg e carriers subject to a non- equilibrated regime, with hole s and ele ctrons each having different mobilitie s. Th is effect was first ob ser ved in 1932 in diamond, and then later in zinc sulfide, and is often called the Dember effe ct. It could be discerned at the limits of materials which were under an intense light. Submitting anthracene to light with wavelength A. = 365 nm, equ al to its fund am ent al abso rption, yields a phototen sion around 0.2 V. The negative 'pole' can be found on the side expo sed to light, as hole s have a greater mobility than electrons, and therefore the former move more readily away from the exposed side. ~ The surface photovoltaic effec t This effect appears at the interface between a metal and a semiconductor and leads to the formation of a potential barrier (a Schottky barrier as detailed in Figure XI-I ). When a metal (M) is placed in contact with a p-t ype semiconductor (SC p), and the metal has a work functi on (W M) which is less than that of the semiconductor (Wscp) i.e. Wsc e > WM, then the equilibration of Fermi levels occurs via the passage of elec trons from the met al to the semiconductor. In the semiconductor, the sa me electrons recombine with hole s and then acceptor ions (A -), negatively charg ed, are no longer stabilised by ne ighb ouring hole s. Th e result is the appearance of a negative space charge du e to a cert ain den sity (N a ) of acc eptors . Given the low value of N j , the spac e charge actually goes quite deepl y into the se mico nductor and the diffusion of electro ns from the metal to the semico nductor stops when the internal field , generated by the space charge directed fro m the met al to semico nductor, equilibrates the diffu sion curre nt. The metal takes on a positive charge near the interface and then holes there observe a barrier !:l W = x. + EG - WM, where X. and EG are the electronic affinity and the gap of the se mico nductor respectively. On the semiconductor side of the interface , hole s movin g toward s the metal ob ser ve

Vacuum level u. ed as reference

Metal

p-type semiconduc tor

Figure XI-I. Representation of the mechanisms giving rise to a phototension in a Schottky jun ction (metal/p-type semiconductor) with W SCp > W M.

XI Organic photovoltaic devices 315 the barrier Wdiff = WSCp - WM. At equilibrium, the modulu s of the current of holes going through ~ W is equal to that going through W diff - The space charge zone is not a great conductor; it is resistive due to the lattice A- ions being relatively immobile. Thus, it is also the seat of any drop or rise in tension if we negatively or positively polarise the semiconductor. If a negative tension is applied to the p-type semiconductor, the current of holes going from the semiconductor to the metal is blocked by a barrier which is raised to Wdiff + IqY-I. If a positive tension is applied to the p-type semiconductor, then the barrier is reduced to Wdiff - IqY+1 and the current of holes from the semiconductor to the metal is reinforced. In effect, we have a rectifying contact which permits the passage of strong currents going from semiconductor to metal and blocks those from metal to semiconductor. Turning now back to the photovoltaic system, there are three possible effect s in the structure of Figure XI-I , in which there is an interface between a p-type semiconductor and a metal where the workfunction of the former is greate r than that of the latter (again, Wscp > WM.): •

•

•

the absorption of light by the metal with the associated generation of a hole of energy that can overcome the barrier ~ W, only when the light is such that hv > ~ W, at the MiSe interface. With a sufficient amount ofenergy, the generated holes can penetrate into the semiconductor and form a phototension at the barrier; with an incident ray of light with hv > EG, an electron-hole pair is created in the hole depleted zone d. The intense electric field, which dominates this zone, of Eint generated on the equilibration of metal and semiconductor Fermi levels, in the direction left to the right of the Figure, efficiently separates photogenerated carriers. The result is a phototension between the metallic electrode and the semiconductor volume; and when the incident wave has a low energy, and a long wave, the optical absorption coefficient is small and in effect there is a poor formation of electron-hole pairs, as hv is not greater than EG . Such waves can penetrate quite deeply into the material and can generate, within the semiconductor volume , electron-hole pairs . In addition, minority carriers can be formed , created at a distance from the depleted zone around their diffusion length (L m ) , to diffuse towards the junction where they are controlled by the field E int . They can then contribute to the phototension at the barrier limits.

Generally speaking the first proce ss is relatively unimportant, and is associated with photoemissions with an intensity of light dependent on the thickness of the metal lic film. When hv > EG, the second and third processes are the most important. Once the photons have an energy much greater than EG, the absorption coefficient is very high and any rays are, practically speaking, absorbed near the interface. A decrease in the photoresponse is thus due to strong recombinations that occur close to the surface, where there are normally a high concentration of defaults . Photovoltaic effect s like tho se described above have been observed in anthracene [Kil 71], tetracene, chlorophyll and phthalocyanine [Gho 74]. We shall detail the photovoltaic effects revealed from these material s in the immediately following Subsection 2.

316

Optoelectronics of molecules and polymers

y The photovoltaic effect produced by an internal field at a hetero- or pn junction As in the case above , the internal electric field across a space charge zone propels in opposite directions the photogenerated charge carriers, resulting in the formation of a phototension. There are now standard texts which well detail the theories associated with devices , essentially made from inorganics, based on these effects (photovoltaic cells) such as Section 9.2.4 of [Mat 98]. In organic materials, most photovoltaic effects are obtained using Schottky pn-junctions, which require two types of semiconductivity. In Section 2 we will cite the example of phthalocyaninejperylene. Here the former component, p-type, is a highly coloured chemical, historically derived from the conversion of disubstituted ben zene derivatives . The second, n-type, is a polycyclic hydrocarbon with the chemical formula C20H12 and is used in the preparation of dyes. To give an overview, the development of solar photovoltaic systems has up until now been essentially shouldered on inorganic semiconductors, in particular polycrystalline silicon. While the maximum yields approach 24%, the cost of fabrication of the raw materials have limited commercial developments. Alternatives using microcrystalline or amorphous silicon have been widely studied and maximum yields have neared 15%. Nevertheless costs remain a problem due to processes based on the use of highly polluting materials, leaving them to compete poorly with other energy sources. In parallel, the use of organic semiconductors, developed during the 1970s and 80s [Sim 85] was also envisaged as a possible route . In the following Sub-sections 2 to 4 we will show the more notable results, before going on in Section II to describe rr-conjugated polymers and organic systems based on specific structures under going development at the present time, notable interpenetrating materials.

2 Initial attempts using organic materials: the phthalocyanines While the first results were hardly very encouraging, with yields of the order of 10- 2 to 10-3 %, significant results, around 1%, were realised in 1978 using merocyanine [Mor 78] . The considerable problem of instability associated with organics put considerable break on their development. This was in part resolved though by using phthalocyanines (Pc) . Indeed, this material has found commercial success as a photoconductor in photocopiers or printers (Xerography). Their chemical structure, shown with the example ofCuPc in FigureXI-2, is close to that of chlorophyll [Keo 98], indicating that real possibilities do exist for the conversion of solar energy. Two clas sical configurations have been explored with this material: Schottky and pn-junctions. In both cases, the Pc layer on being submitted to light rays is used to generate electronhole pairs . Their separation by an internal tension, which governs the limits of the junction, can give rise to a photocurrent in an external circuit. As shown in Figure XI-3, the Schottky junction is typically prepared with a sandwich structure. The p-type semiconductor, phthalocyanine (see also Chapter X, Section IV-I-a), is placed between an electrode that ensures an ohmic contact (Au or ITO , which have high work functions) and a second, rectifying electrode which permits the passage of current from the p-type semiconductor towards the latter metal (AI or In, which have low work functions) .

XI Organic photovoltaic devices

317

Figure XI-2. Structure of the CuPc molecule. IT O or Au

Pc

Figure XI-3. Configuration, under light, of a Schottky PclAl or In j unction.

The rectifying effect, detailed in Figure XI-4, can only be efficient if the phthalocyanine layer exhibits a negative space charge and is truly p-type. If the layer is deposited under air it shows strong rectifying properties (Figure XI-4-a), however, if deposited under vacuum the same properties are only weakly exhibited

I I I

I

- - --

~

v

(at

Under light (c) _ . /

.-

.

'

Figure XI-4. The (a) rectifying or (b) non-rectifyi ng behaviour with respect to deposition conditions used with CuPc; and (c) the resulting J(V) curve obtained on exposure to light.

3 I8

Optoelectronic s of molecule s and polymers

(Figure XI-4-b) . In general, the degree of contamination by oxygen is highly variable, making it difficult to compare result s from different laboratories. However, the characteristic curve obt ained under light, as in Figure XI-4- c, can be used to determine the open circuit tension (Vco) (the photovoltaic regime) and the short circuit curre nt (l ee) . The maximum electrical power (P maxJ is such that P max = Vmax Jmax = Jcc Vco F. The factor F, called the fill factor , dep end s on the series resistance of the device. Vmax and J max are defined in Figure XI-4. The yield of the system, generally expressed as a percentage, is the ratio of maximum electrical power to the power of the total incident light. Using this Schottky configuration , yields of I% were finally attained using materials based on a spacer group (X) and non-metallic phth aloc yanines (X-HzPc) and lighting through an indium electrode. The poor tran sparency of the electrode resulted in a global efficiency of only 0.2 to 0.3%. Lighting the device from the side of the tran sparent electrode (ITO ) did not improve the result due to the Pc layer filtering light, which otherwise would have reached the space charge zone situated near the indium electrode. The use of pn-junctions should, in part, help overcome the problem associated with absorption due to the metallic electrode, used as a rectify ing contact. This is because the space charge now appe ars at the interface betw een the Fl- and p-type semico nductors. Structures based on n-type perylene and p-type Pc, schematised in Figure XI-5 , have been experimented with. Better perform ances than those observed with Schottky structures were obtained, due to a higher degree of light tran smission to the space charge zone. Exciton s which diffused up to the inte rface between the two semiconductors were thu s ionised and the electrons and holes could be coIIected. With devices sufficiently thin , a fill factor of 0.65 can be obtained. Tang, using this meth od [tan 86], reali sed conversions of the order of 0.4 to I%. However, a higher value is required for the sys tem to be viable at an indu strial level.

3 Solar cells based on pentacene doped with iodine [ScM OOb] In order to dope pentacene with iodine, the pentacene is introduced into a hexane or acetonitrile solution of iodine and left there for 72 h. Thi s proce ss gives rise

Figure XI-So Organic pn-jun ction under light.

XI Organic photovoltaic devices

319

to a good p-type semiconductor. The iodine molecules are incorporated between layers of pentacene molecules and the resulting charge concentration can vary from 107 to 1018 cm- 3 . The iodine enables a high charge mobility (comparable to naphthalene and anthracene at 1 to 2cm 2 V-I s-I) and good electrical conductivity, in addition to the efficient separation of charges. Useful also is the displacement of the absorption spectrum (threshold at 1.4eV) towards that of the solar spectrum. The increase over several orders of size of the quantum yield of this system, which can reach from 1.9 to 2.4%, can be reasoned by the following mechanism: • •

the incident photon is absorbed by the active zone where an exciton is formed (Figure XI-6~a); the exciton diffuses into the neighbourhood of an iodine dopant molecule to generate an excited state at the pentacene-iodine-pentacene complex level (Figure XI-6-b);

000000 000000 000000 000000

000000 000000 000000 000000

::~O~:

OOi~/)OO

~

OOOO

() 7)000 u. o 0000 000 000000

000000 OOOOr 00000 00000

0 0:0r;,g0

~

o 0000 o 0000 o 000

000000 4 --

(a)

000 00 000 '-f}0 000 00 000000

(b)

000000 o0fIJO (). 000 000 000 0 000 0

(d)

(e)

OO~OO

000 00 000 '-f}0 000 00 000000

-\0:

Figure XI-6. Charge photogeneration in the pentacene-iodine system: (a) photogeneration of an exciton; (b) diffusion ofthe exciton near to the iodine ; (c) transfer of negative charge towards the iodine and dissociation of the exciton ; and (d) separation of positive and negative charges under the field (E) effect.

320 Optoelectronics of molecules and polymers • •

the negative charge of the exciton is partially transferred towards the doping molecule (Figure XI-6-c) and thus aids in the dissociation of the exciton; and the hole and electron are finally separated and the charges carriers are collected, with the help of an electric field, to yield the photocurrent. (The electric field which gives rise to the collection of these charges is an internal field generated by a Schottky junction or a heterojunction).

The lifetimes of such devices remain controversial and it is for this reason that any results announced should be viewed with some caution. Cells with dimensions 5 by 5 mm have been made using the structure ITO/iodine doped pentacene (50!tm thickj /Mg or Al and have yielded Yeo = 995 mY, lee = 4.5 rnA cm- 2 , and a fill factor (ff) of 0.45 .

4 The general principle of Graetzel and current organic solar cells In general terms , and as we have already seen in some examples, the heart of a solar cell consists of a semiconductor which normally absorbs photons and thus passes electrons from valence to conduction band s. This requires that the photons have at least a minimum energy and that excess energy is lost, even if this means that in the best materials 50% of the available energy is wasted. It is for this reason that the size of the forbidden band (Ea) must be well adapted to the incident light. Excess photon energy, with respect to Ea, results in the thermal excitation of photoelectrons towards the LUMO band minimum (with, for example, an excitation of lattice vibration energy). In the natural world, green plants which use solar energy are apriori subject to the same problem. Graetzel [Gra 91 J was inspired to transfer the solution that they provided to photovoltaic system s. In plants , light is recovered by chlorophyll molecules which exhibit an absorption band well adapted to the solar emission spectrum. In effect, few photons are not used. Once the solar energy is absorbed, it is transferred via a protein complex in which a separation of charges gives rise to chemical reactions. These reactions convert carbon dioxide and water into hydrocarbons and oxygen . In terms of photovoltaic systems , the principal idea applied during the I970s came from the same logic. In order to limit losses, the processes of light absorption and conversion to electrons were separated. Initially therefore, the structures were based on a layer of absorbent dye molecules deposited on a SC surface . Once the dye molecules were excited, they liberated, or rather transferred, an electron to the SC conduction band . This completely removed the dependence of photon absorption with respect to the SC absorption spectrum. The problems that are now faced are : • •

the efficiency of electron-hole pair (exciton) disassociation; and the efficiency of charge transfer towards the SC and the following collection of charges at the electrodes.

In practical terms, the slow transfer of charges from the dye required that this layer be extremely thin. If not, numerous recombinations could occur. The available material

XI Organic photovoltaic devices

321

for absorption is thus reduced and the numb er of electrons that can be generated becomes limited. So, rather than using a specifically semico nducting material, such as crystalline Ti02, Graet zel prep ared a porous material from a large number of colloidal Ti02 particles and impreg nated it with dye molecules . The particles, whic h were hardly 20 nm across, were in effect inserted within a dye layer of about I micron . The effec tive spec ific surface area of the dye using this technique hence became very large indeed. Now standardised, the Graetzel cell is made up of a liquid electro lyte sandwiched between two transparent electrodes.The instability ofthe system, due to evaporation of the electrolyte, however, does demand a low operating temperatu re range, something which has tended to limit their wides pread use. Poly(para-phenylene vinylene) (PPY) and its derivatives are, at present , materials widel y under investigation for insertion in porous Ti02 matrices. A problem remain s in the macromolecular size of the PPYs, especiall y important now that the Ti02 is prepared on a rotating disc from a very fine powder. At the Universities in Santa Barbara and Cambridge, principally, research has now centred on the use of buckm inster fullerene (C60) in place of Ti02 in composites with PPY. Here a PPY chain is grafted onto C60, and we will look at these materials now in more depth .

II n-Conjugated materials under development for the conversion of solar energy There are many envisaged advantages to maki ng polymer ('plastic ') based photovoltaic systems. They includ e their ease of fabrication and manipulation, their flexibility and low weight and their low cost, all of which are good reasons to imagine their facile integrati on into buildings. These expec tations have done much to push research rapidl y forward over the last few years.

1 Metal-Insulator-Metal structures On using the structure of organic LEDs as a basis, the classic metal-insulat or-metal (MIM) structure was envisaged as a possible route to the preparation of photodetectors. Organic LED s were , incidentally, the first optoelectronic devices made using n-conjugated polymers. As shown in Figure XI-7, once a cont act is made , then the alignment of Fermi levels adju sts band positions. Without an applied tension, i.e. Y = 0, there is a swing in the polymer bands so that, still without the application of the external field , there appears an external field Eint = - grad Yint =1= O. The difference in intern al potential is due to the energy differen ce D. Wint = Wanode - Wcathode, i.e. the difference in work functions of the anode and cathode metals. With the inception of a photon an exci ton is produ ced. Under the photovoltaic regime, only the slight difference in internal potenti al controls-with a low yieldthe dissociation of the photogenerated quasi-particle. In practical terms, the generated phototension is not great enough to be of any use. With the struct ure ITO/ PPY ( 120 nm

322

Optoelectronic s of molecules and polymers

WlUlOd<

Figure XI-7. Photogeneration of charges in a MIM structure.

thick j/Mg, the tension at open circuit is 1.2 V and the short-circuit current is equivalent to a quantum yield of I% (electrons collected per incident photon). This yield, however, improves drastically once the system is subject to an inverse polari sation (around IOV) and can permits envisaging this system applicable as a large surface photodetector. The use of a Schottky structure resulted in only a slight improvement in charge photogeneration. It is generally acknowledged [Mar 94] that charge photogeneration in the volume of the material occurs through dissociation of exciton singlets under field effect s. Instead of traver sing the polymer layer, electrons are either trapped or undergo combination with hole s, giving rise to an rather unremarkable result.

2 How bilayer hetero-structures work and their limits Following on from the fund ament al principles, deri ved from the photo synthetic systems described in Section I, any impro vements in the photoph ysics of the precedin g organic monola yer structures (MIM or Schottky M-SC) necessitate the use of bilayer systems. The use of a donor/ acceptor structure would allow the efficient transfer of photo-electrons from a 'donor' semiconductor n-conjugated polymer (p-type semiconductor POD ) toward s an ' acceptor' material which can either be an n-type rr-conjugated polymer (which are rather rare ), or an accepting molecule, of which the archetype is C60 .

a Production and collection of photoinduced charges The making and harve sting of photoinduced charges follow s the proce sses given below: • •

the incident light generates electron-hole pairs at the p-type donor polymer (POD) due to electron transition s from n HOMO to rr" LUMO band s; the created pair is separated close to the acceptor, and electron s generated in the POD LUMO band are accepted on energetically favourable term s into the acceptor LUMO band. As Figures XI-8 (which shows the flat band co ndition) and XI-IO

XI Organic photovoItaic devices (b)

(a)

1t*MEH-PPV

-LUMOC .............

2.8 1t*MEH-PPV LUMO-C~

..............EF

3.7

323

0

.

4.3 Al 4.7

1tMEH-PP

ITO

5 1tMEH-PPV HONIOL60- .1

--

- Eint +

Figure XI-S. (a) Flat band regime of MEH-PPY jC60 system; (b) Energy levels in the MEHPPYjC60 system in the shortcircuitstate.

•

indicates , this is due to the acceptor LUMO being of a lower energy than that of the POD LUMO ; the separated charges are propelled in opposing directions due to the internal field-to be collected-and as schematised in Figure XI-8-b.

b Transport of photoinduced charges to the electrodes In order to transport charges up to the electrodes, the photovoltaic system exploits the relative conductivities of holes and electrons with respect to the constituent donor and acceptor molecules in the composite. a Transport in the p-type donor polymer Once the donor p-type rr-conjugated semiconductor polymer (POD) is excited by incident light, the first process it undergoes is the transfer of electrons from the bonding rt-orbital to the antibonding rt" -orbital. Once the excited electron is transferred to an acceptor molecule, in this example C60, then a polaronic state, associated with a positive polaron , appears on the POD chain. What we are observing here, in a sense, is a localised photodoping. The polarons have transport mechanisms detailed in Chapter IV. ~ Transport in the n-type acceptor C60 Once an electron is transferred to the C60 from the POD, it can benefit from the n-type semiconductor character which C60 exhibits to be transported throughout the C60 phase. Chapter III gives more relevant details .

y Overall result The rapid transfer of a photoinduced electron from POD to C60 permits reinforcement of the generated charges at POD and assures the separation of photogenerated charges-by enhancing their stability through a polaronic effect. It is thus possible to obtain a heterojunction with a rectification factor of the order of 104 for a bilayer structure based on poly(2-methoxy-5(2'-ethyl-hexyloxy)p-phenylenevinylene) (MEH-PPV) and C60, as schematised in Figure XI-8-b. In addition to which, it is possible to yield either a photocurrent or a photovoItage .

324

Optoelectronics of molecules and polymers

c Properties and practical limits of donor-acceptor bilayer systems The photophysics of bilayer systems have been studied in depth by C.J . Brabec and N.S. Saricifti [bra Ola] . Notably, it was shown that the time required for a decrease in the luminescence of MEH-PPV (1: = 550 s) was reduced by more than one order by the presence of C60 . Thi s 'quenching' of luminescence by C 60 seems to be a general phenomena for all rr-conju gated non-de generate polymers. In particul ar, it indicates that the transfer of a photoinduced electron occurs during an interval sufficiently short to limit any radiative relaxations of the excited, conjugated polymer. However, a bilayer system must present some limit s in performance, as even though the system has a high optical absorption coefficient (~ 1 05 cm""), the absorption zone is considerably greater than the exciton diffu sion length. As the exciton can only dissociate in a volume close to its zone of existence (the region at the interface between the donor polymer and the C60 and no further from the C60 than the short exciton diffusion length), all photoexcitations generated outside of this zone are condemned to undergo recombinations without generating efficiently separated photo-carriers. They cannot, for example, participate in the form ation of a photocurrent. Also, the photovoltaic effect arises only from photons absorbed with this thin layer. An improvement in this situation can be made by introducing layers with high levels of optical absorptions or, alternatively, by using crystalline semiconductors of a high level of purity which allow longer exciton diffusion lengths. Another approach to impr oving this system involves using elaborate thin films constituted of intimate mixture s of both the conjugated polymer and C 60. There is thus a large surface area for absorpti on which is alwa ys clo se to a POD and C 60 interface , where the dissociation of an exciton can be assured [Hut 011. Figure XI-9

------------------------ r-

11 :;

0

r c--

,........... t'!l

l<

e.

0 Q

:l

., Q

N Q

"Cl

:l

::r

'-

J

~

'"

'--

S'

:l

e,

'" ~ o· :l

'--

"Cl

-

"Cl

'" ::

> n n

'"

S' ., ::r

e1

o ~

::r

8-

'--

'"

-

(a)

(b)

Figure XI-9. Schematisation of (a) a bilayer structure; and (b) an interpenetrating structure.

XI Organic photovoltaic devices

325

gives two repre sentations corresponding to a classic bilayer device (Figure XI-9-a) and a device made using a compo site of two interpenetrating phases (Figure XI-9-b ).

3 Volume heterojunctions a Principle At each donor-accept or element constituting the composite, the same mechanism is reprodu ced. That is, the creation of the exciton, its disassociation and the transfer of an electron from the donor to the acceptor, and the eventual separation- in opposite directions-i-of the hole in the donor phase and the electron in the acceptor phase under the influence of the internal field generated by the asymmetric electrode s. This process is detailed Figure XI- IO. As the zone of interaction betwe en the donor and the acceptor are not reduced to a simple plane (as found in the bilayer system in Figure XI-9-a), these compo site devices present a zone for the absorption of light which is sufficiently thick while at the same time maintaining the process of dissociation and separation of charge carriers in the volume of the layer. Outside of this , the transport mechanisms are the same as those exhibited by the bilayer device, and the mechanism used in the physical transfer of charges is the same as that schematised in Figure XI- I I. It is for these reasons that an interpenetrating composite can exhibit high (close to 100%) efficiencies in charge generation. It can be thought of as interpenetrating latticework of donor/ acceptor heterojunction s, allowing efficient separation of photogenerated charges throughout its volume via the two n- and p-type phases. In effect, the photoactive layer is one mass heterojunction. The two intermixed material s of different gaps give rise to zones of sub-systems with levels where electrons are transferred between donors and acce ptors. The ideal situation is that the acce ptor is placed at a distance less that the diffusion length of the exciton so that the separat ion of the photogenerated pair, or in other term s the dissociation of the exciton, is facilitated.

b The results Under solar light, efficiencies have reached 3% [Brab Ola]. The composition and morphology of the film is, however, a critical factor. Rather than obtaining composites which are completely homo geneous, the films are rather disorgan ised, with isles form ing of each component material. The size of these aggregates depends on the solvent used to prepare the film. As an example we can site the use of chlorobenzene and toluen e, the former giving rise to smaller aggregations. Th is effect acts directl y upon the performanc es of the device as the smaller the size of the aggregates, the greater the specific interface zone available for dissociation of excitons. We should also remember that the donor-accept or matrices should interpen etrate to a sufficient degree, so that the transport of the two charge carriers is assured. A high concentration of C 60 can increase both the yield of photo-carriers and their lifetime s, resulting in a global increase in photo-current.

HOMO

Donors

eo

_

I .

.....

Donor

L UMO ___.

1I0MO

HOMO

L UMO

e e ti31:

- en '" :: C :"

i:~

Excited donor

'-"

.....

•

--

c ..

1I0MO

Accep tor

':xcit:ttioll

LUMO

HOMO

Donor+

e-e

L UMO - - - -

re laxat ion

---I--

i

C'

'-'

__

Acceptor-

Dnnor'"

-

-

-

-

-----:.i---

HOMO

HOMO

LUMO

I

Figure XI-IO. Schematisation of: (a) the formation of an exciton at an irradiated donor; and (b) transfer of an electron from an excited donor to an acceptor. In effect, a hole or electron carrier are formed , respectively, from the donor and from the acceptor.

EJI

I

B

LUl\1O

tv

W

'"

« ~

o

'0

::l 0-

'" t.:>

o 0" n s:: 0"

3

'" o ....,

::l

"'.

g

[

~

o

0\

XI Organic photovoltaic devices

~ o

n-conjugated polymer chain (POD)

Cathode AI

327

e

Cw Charges arc photo-generated Photo-induced electrons and transported

Positive polarons transported along POD.

Figure XI-H. Schematisation of the physical processwithin a composite of POD and C60.

Outside of the system based on MEH-PPV/ C60, the research teams of A. Heeger at Santa Barbara [Yu 95] and R.H. Friend at Cambridge [Hal 95] successfully experimented, independently, with a composite based entirely on polymers. The system used was based on MEH-PPV and CN-PPV, the former acting as the donor and the latter, a cyanide derivative of MEH-PPV, acting as the acceptor. CN-PPV exhibits a high electron affinity due to the presence of -C == N groups substituted on its backbone (which have a character discussed and detailed, respectively, in Chapter X, Section IV-I-c and Appendix 1-10). As shown in Figure XI-8-a , and much like C60, the LUMO of CN-PPV is below that of MEH-PPV. More recent studies have resulted in even higher efficiencies by gradating the stoichiometries of the different components in the layers, facilitating conduction of electrons and holes in the volume of the heterojunction [Bra Olb] . c Using n-conjugated polymers grafted onto C60 One way of reinforcing the contact between donor and acceptor molecules, and overcoming phase separation, is to directly graft the donor chain directly onto the acceptor. Thus , a bicontinuous medium is obtained. There are two notable examples: • •

grafting poly(para-phenylene) onto C60 (a route chosen by E. Mignard et at) [Mig 01]; and grafting oligo(para-phenylene vinylene) (OPV) onto C60 [Nie 99].

Using the structure ITO/C60-0PV/Al allowed both the generation of electrons and holes and also their collection at opposing electrodes. On exposure to light , using the oligomer tri(para-phenylene vinylene) (3PV) an open circuit tension of 0.46 V was obtained, roughly equivalent to the difference in work functions of the ITO and AI. Under short-circuit conditions, the photo-current was found to be 10 I!A cm- 2 and the fill factor 0.3. The rather poor yield, nevertheless more than two orders greater than that of a layer of 3PV alone without the C60, could be explained by considering that the transfer of photoinduced electrons (Figure XI-12, level I) between the 3PV and C60

328

Optoelectronics of molecules and polymers C60-\3PV)

_0#----.. . . . . . --

+

(C60H3PV) ~ ---I I

I

!

II

1(C60)-3PV

I I I

hv

I

! I I

3(C60)-3PV

.....L._ _""""'-'L-

C6o-3PV

-'-_-:fc::.un::;-damental level

Figure XI-12. Schematisation of theenergylevelsinvolved in the relaxation of C60-3PV (after I.E Nierengarten, G. Hadziioannou, MaterialsToday, 4, (200I), 16).

Coo-4PV

Figure XI-13. Representation of the compound C60-4PV.

molecules was in competition with an energy transfer to the lowest energy singlet state of C60 (Figure XI-12, level II). A better result was obtained using 4PV in the structure shown in Figure XI-B, indicating the relationship between structures and characteristics. These results do highlight the route towards other developments based on, for example, new derivatives of C60 [Nie 0 I].

III Additional informations about photovoltaic cells and organic components 1 Discussion about mechanisms leading to the generation of charge carriers in organics The making and harvesting of photoinduced charges follows the processes given below where donor and acceptor materials exhibit an interface which induces charge separation. Figure XI-14 details the various processes of varying efficiencies (11), which are: (1) photon absorption (l1A) . The incident light generates electron-hole pairs at the p-type donor material (POD) due to electron transitions from n HOMO to n" LUMO bands . llA depends on the value of the optical absorption coefficient and on the thickness of the donor material;

XI Organic photovoltaic devices 329

Anode (6},

iJ

I

'\. , 0 " \. - "(J) »TJ'; -; , O~ nee '0 ,HOMO I

;:.

I

\.

I

\.

Figure XI·14. Mechanisms of photocarriergeneration in organics.

(2) generation of excitons.The generation of an electron-hole pair, by photoexcitation, results in an excited but neutral state with a limited, finite lifetime. This state is termed an exciton and consists of an electron and a hole paired by excited energy states (Eex) within the limits of permitted bands (LUMO and HOMO bands, respectively) . The occupation of these excited states , the LUMO by the electron, the HOMO by the hole , is termed a non-recombined exciton generally observed in organic materials; (3) exciton diffusion (lldiff). This quasi-particle diffuses inside the donor material as long as recombination processes (of the hole-electron pair which make up the exciton) do not take place. The Forster (long range) or Dexter (between adjacent molecules) transfers can take place between an excited molecule (considered as excitation donor) and a molecule that receives the excitation (excitation acceptor) . TJdiff < I because various recombinations can occur in the thick donor material ; (4) hole - electron separation (exciton dissociation) (nrc). If the diffusion length is sufficiently long so that the exciton meets an internal field, hole and electron separation occurs. The internal field may be obtained at a donor-acceptor interface, provided the LUMO level of the acceptor is lower than the excitonic state located at the bottom of the conduction band of the donor. More precisely, Figure XI-14 indicated that the condition which must be fulfilled is Eex > Ipd - XEa, where Eex is the bounding energy of the exciton, Ipd is the ionisation energy of the acceptor, and XEa is the electronic affinity of the acceptor. As this condition is generally verified, nrc ;::;; 1; (5) carrier transport towards the electrodes (lltr). Thi s transport involves the classic mechanism for hopping proce sses in organic materials. Traps can reduce the mobility. If carriers are not infinitely trapped (as for example in lattice defects such as dislocations), we can consider that lltr = I; and (6) charge collection at the respective electrodes (nee) - For this to occur, (EF)eathode < (ELUMO)aeeeptor and (EF)anode > (EHOMO)donor. When fulfilled, nee ;::;; I.

330

Optoelectronics of molecules and polymers

2 Electric circuit based on an irradiated pn-junction; photovoltaic parameters For this study, we will consider a pn-junctionwith an ohmic metallic contact.

a Current inside the junction under short-circuit conditions and in the dark The Fermi levelsare alignedon the samehorizontal line and thebarrierwhichprevents the major carrier flowing is qVbO = WsP - WsN, where WsP and WsN are the work functions of the p- and n-type semiconductors (Figure XI-IS-a). If we denote with the index 0 the currents in the dark, the resulting electron and hole currents are, respectively: JOn = l onN JOnP _= 0 (~oduli of the majority and minority electron currents are equal); and jop = Jopp + jOpN = 0 (moduliof the majorityand minorityhole currents are equal).

±

(a)

-

J""

h laj

P -

(c) qV ~M

M,

E. -_J__________________________ E,.~ _ ----'--.L..J qV oc - = q(V

-

Vb)

bO

-_---J z~

}-

-

Figure XI-IS. Current densities in a pn-junct ion (a) under short-circuit condition s and in the dark; (b) under short-circuit conditions and under irradiation; and (c) in open circuit and under irradiation .

XI Organic photovoltaic devices

331

The resulting current is of course zero: Io = Ion + Iop = 0, and we can write: Io = cJOnN + l opp ) + UOnP + IOpN) = l Maj + Is, with (10nN + l opp) = l Maj (current of majority carriers) and UOnP +IOpN) = Is (saturation current).

b Current inside junction in short-circuit and under irradiation Under short-circuit conditions, the Fermi levels remain aligned, but in the irradiated areas the ~ermi le~el s b~come £he qua~i Fermi levels as shown in Figure XI-15-b. We then have JMaj = JnN + JpP :::::: JOnN + Jopp because the irradiation does not drastically modify the density of the majority carriers. Conversely though, the minority carrier densities are strongly modified in the n- and p-regions because of the irradiation which generates the hole-electron pairs. The minority electron and hole currents are thus jnP » jonP and jpN » jOpNand the resulting current is practically generated only by minority carriers as in Is = InP +IpN. Inside the space charge area there is some recombination of electrons and holes to give a recombination current which we will neglect. In addition, hole-electron pairs are also generated in the space charge area and giving a current going from the n to the p region which is denoted JGen in Figure 24-b. As a result, the irradiation Erodu: es a reverse current (going from the n to the p zone) and we can write: lph = jsat

+ JGen·

c Junction in open-circuit and under irradiation (Figure XI· IS-c) This situation can be obtained when the junction is connected to an electrometer/vo ltmeter of high impedance (Zelectrometer ~ 00) . The Fermi levels move closer to one another as the metallic electrodes are not directly connected by means of a short-circuit and a voltage appears across electrodes. Becau~e the : ircuit i: open, the current density is zero and 1 = l Maj + Isat + l Gen = 0, where jsat + JGen = Jph is the same reverse current as that calculated above in §IV,2,b. l.:,he resul~ng cu_rrent bei~ g zero, the majority current must now be increased so that IJMaj I = Usat + JGenI = IJph I· In fact, if the majority current is increased, then that means that the barrier which prevented the flow of the majority carriers is reduced from its initial value of qVbOto qVb, which verifies qVb < qVbO. So that the barrier observed by majority carriers becomes qv j, (for example, barrier to electrons on the metal (M2) side), the Fermi level must be raised on the same side by a value qVbo - qVb = qVOC, and thus Voc = VbO - Vb is the voltage which appears at the contacts in the open-circuit configuration. The n region which is initially positive with respect to the p region with a value VbO, becomes less positive under irradiation (with a value Vb < VbO). What is important is to underline that, contrary to what is often stated, the voltage in an open, irradiated circuit is not equal to the voltage difference between the surfaces

332

Optoelectronics of molecules and polymers

of nand p regions and (Wsp - WsN)/q = VbO. This value is the bias voltage which has given, not without some error, the name photovoltaic. As demonstrated above, the value for Voc is given by the variation in the potential bias between its value without and with irradiation and it is only when Vb -+ 0 that we have Voc ~ VbO. VbO = (Wsp - WsN)/q is thus the maximum value that can be reached by Voc. the open-circuit voltage for an irradiated pn-junction.

d Electrical characteristics If the non-irradiated junction is biased with a voltage V, the current of the minority carriers is unchanged at Js- while the majority current is exponentionally reinforced to become JMaj

= is exp (~~) . The resulting current density is thus:

When the junction is irradiated, the current density becomes:

(1)

and we have the characteristics shown in Figure 25-a . Under irradi ation , the curve no longer goes through the origin. The irradiated biased junction can thus work as: -

a rectifier with a forward bias V > Voc in the first quadrant;

-

a photodiode with a reverse bias in the yd quadrant where V

«

+ Jph) that is to say with js Jph, J ~ -Jph. The measured current is approximately proportional to the irradiation and when V -+ 0, I -+ Icc ~ - Iph (short-circuit current); and a photovoltaic cell without external bias, but with a current flowing through a load resistance RL in the 4th quadrant so that the photovoltaic function is not reduced to the open circuit only. In this 4th quadrant, the product of VI is negative so that power (and electrical energy) can be produced by the device . With I' = -I, the curve I' (V) can be depicted for the general case in the first quadrant (see Figure XI-16-b) where the optimised (maximum) power (P max) fits the values I' = I:n and V = V m so that P max = Vml m. J = -(js

-

kT « 0 and IV I » -, q

The fill factor (FF) is then defined as the ratio of P max to Voc x Icc and can be P max

written FF = - - - Voc x Icc

XI Organic photovoltaicdevices 333

(b)

(a)

J

l'

1ICClr_ I;" .-: ~·-.-: ~·-.-; ~·-.-; ~I · Ma~i!Jl)J~ : .:

: l:io~~r: .:.:.:~

------:-l-==--+--+V

·rectangle .:' } •• •• •••••••. ••.L ·

.

.•

.

..

-I

v; Voe ee =-

ph

Figure XI-16. (a) J(V) characteristics; and (b) I' (V) characteristics with the maximum power rectangle.

Furthermore, the open-circuit voltage (Vod can be directly deduced from relation (1), where J =0, so that we have:

kT Voe=-Log q

(J

ph kT (Iph -+1 ) = -Log -+1 ) . Js

Is

q

(2)

e PhotovoItaic parameters The internal quantum efficiency (IQE) , is defined by IQE = Jldiff x T]Te x T]tr x nee . while the external quantum efficiency (EQE), is given by EQE = T]A x IQE . In effect, the EQE, obtained by the multiplication of all the efficiencies, represents the ratio between the number of generated electrons to the number of incident photons. This coefficient, also named the Incident Photon-to-Current Efficiency (lPCE) represents the ratio between: -

-

the measured photo current ex~ressed as the number of collected electrons in the where Icc = Icc/S is the current den sity unit time and per unit area: lc~ S = in A cm - 2 ; to the intensity of the incoming mon ochromatic light expre ssed as the number of incoming photons (of "'- wavelength) per unit time and area : ~~j~ = ~; cPe is the incident energetic flux (expre ssed in Watt) and E, is the illumination defined by E, = cPe/S (in W / cm 2 ) . Thu s, we have:

¥,

EQE = Icc x hc = Jee x hc .

cPe

e"'-

Ee

e),

With "'- expressed in urn , numerically, EQE = 1.24 {;~e . Finall y, the external power conversion efficiency T] is defined as the ratio of the maximum electric power P max to the energetic flux cPe = E, x S i.e.T] = = XVoc. This yield is the maximum value for the theoretical yield , and to optimi se FFxlc;e

P;:,

this yield we have to optimise each of the three factors in the numerator. In practice,

334

Optoelectronics of molecules and polymers

the yield is diminished by two factors: the effect of the serie s and shunt resistance s and the effect of reflection at the surface of the cell.

3 Circuit equivalent to a solar cell a Equivalent circuit for an ideal solar cell As I = Is [exp

(~~)

- I] - Iph, the biased and irradiated pn-junction can be seen as

a current source Iph (a reverse current proportional to the incident light) in parallel with a diode in the dark which delivers the current i = Is [exp ( ~~) -I]. We thus obtain the equivalent circuit for an ideal solar cell, outlined in Figure XI-17, which consists of an ideal diode and a current source in parallel. If the junction is connected to a load resistance (Rd , the voltage (V) is the result of the ohmic fall of the current through RL (in open circuit, RL -+ 00 and V -+ Voc). The orientation of Iph towards RL produce s a V which induce s a forward bias across the junction so that the current (i) is a forward current in the direction opposite to Iph . Altern atively, we can consider that the direction of the photocurrent in the load resistance induce s a voltage across the j unction which generates the i forward current in the direction opposite to Iph , so that in the external circuit we do not observe the entire Iph photocurrent, but only I' = Iph - i.

b Equivalent circuit for a real solar cell When the contact resistances (electrode resistivity and metal-materials interfaces) and the ohmic losses (due to the bulk resistivity ofthe material s) generate a non-negligible resistance in relation to the load resistance , we must include in the equivalent circuit a series resistance (rs), as detailed in Figure XI-18. If we denote Vj the voltage across the junction, the voltage V across the cell is reduced to V = Vj - rsl', and in the first quadrant we have: j) (qV + rsl'] )- ] I, = Iph - Is [exp kT - I ] = Iph - Is [ exp ( q[v kT I .

h\O, (a)

(b) ! " p

-

r,

t

- I

ts[exp(~H

If = I

ph-i L

Figure XI-17. Electrical circuit of (a) ideal irradiated pn-junction connected with a load resistance RL; and (b) equivalent circuit.

XI Organic photovoltaic devices

335

L

Figure XI-IS. Equivalent circuit for real irradiated pn-junction with (a) series (rs) resistance; and (b) with series (rs) and shunt (rp) resistances.

Furthermore, when leakage currents (which include shunt currents through short-circuits) arise across the cell, we can take into account this component by introducing a parallel resistance (rp) such that rp --+ 00 when leakage current ip ~ 0. We then have: I' = Iph - i - i p = Iph - Is [ex p

(:i)-I] - ~:

Repla cing Vj by Vj = V + rsI' we have: ( q v + rsI' )- ] V +rsI' I - -I, = Iph - Is [exp kT rp

(3)

While for the ideal cell in the external circuit we do not find the total photocurrent (Iph) but only I' = Iph - i, in a real cell the same reduction is more pronoun ced. This is because the shunt resistor Ip introduces the linkage curre nt ip leading to I' = Iph - i - ip. Simultaneously, series resistor consumes the voltage rsI' . In ideal cells, we have r, = and rp --+ 00 : so, the departures from these values give an evaluation of the imperfections of the diode; considering that rs always presents a small value and that r, « f p, we can then estimate the values r, and f p: by differenti ation of expression deduced from (3):

°

( qv+rsI' ) f( I, , V) = Iph-IS [exp kT - ] 1 - V+r rsI' · - I, = 0, p we have:

~ir =

-f, that leads to (with v

dV ] [ dI' 1'-0 -

~

-rs

_

UT

= kT / q ~ 0.024 V):

rp Isrp V 1+ exp UT

(because when I' = 0, V = Voc dV ] [ dI' v=o

(as rs is small)

~ - rs -

»

UT )

~

Isrp rsI' I +-exp UT kT

UT

~

- rs

336

Optoelectronics of molecules and polymers

ideal chara1e~ stic

al characteristic

f

I

/

;.- /--------. / ~

I

- 1 (.d

/

fs

;1

-- - - -::A..

I

ea case:

- I ~-d-~O'\ f s ~ oo an f s

=

v

---

-'--'~:'-:==--------------A~_ _~ Icc

-I (.d al -1 d '\ rp ; I e case : rp =0 an rp~oo

Figure XI-19. Reverse values of series resistance(r;: I) and shuntresistance (rp1) linkedwith the characteristics slope at V = V oe and at V = 0, respectively.

With I = -I', we also have r; 1 = [1~ ]1=0 and rp = [1nv=0' The slopes of the characteristic I(V) under irradiation, calculated for I = 0 (at open circuit) and in V = 0 (at short circuit) give the inverse values of the series resistance rs and of the shunt resistance rp, respectively (see Figure XI-19 ). If r, incre ases, the slope of the characteristic decreases in the first quadrant and Icc also decreases. If rp decreases, the slope increases in the fourth quadrant, and V oe then decreases. l

Note: To take into account a more general behaviour of the current inside the real jun ction (recombination current in the space charge area ), eqn ( I) must be written

) -I]

I

in the form : J = js [e xp (~~4 -Jph, where n is the "ideality factor " (n = in the ideal case). The relationship between Voc and Iph expre ssed in eqn (2) obtained when rs = 0 and rp ---+ 00 , is thus changed to Voc = also modified and becomes

I (I

+ r~) - ~ r p

p

= Is [exp

(~(V nkT

n~T Log

rsl)) -

Cr;+I).Eqn (3) is

I] - Iph .

4 Possible limits

It can be thought that a power conversion efficiency as high as 10% can be reached with organic solar cell s ([Nun 02] and [Peu 03]). Given that

l]

=

FF x l ee x Voe ~e

e

Jcc

= FF x Voc x Ee

= FF x Voc x - x EQE , hv

XI Organic photovoltaic devices

337

then (ef § IV, 2-e): eVoe II = FF x - - x llA x IQE hv

= FF

eVoe x -hv

X

llA x lldiff

X

nrc x lltr x nee-

As nrc . lltr and nee are all close to I (see §IV,I), we can consider the power conversion efficiency (ll) as being essentially the product of 4 terms as in II = FF X (qVoejhv) x llA x lldiff. With small series and shunt resistances (r, < 50 Q andr, > 25 kQ), and FF ~ I (see §IV, 3), we have for the power conversion efficiency. II = e~~c X llA x lldiff. With e~~ ~ o;/e~v = 0.25, llA = 0.5 and Ildiff ~ I, a value we can hope to achieve with appropriate systems [Nun 02], we finally have II ~ 10%. So, with at least 3% power conversion efficiency already demonstrated, and with improvements to reach 10% in the years to come , "plastic solar cells" may significantly contribute to future energy needs .

5 Examples; routes under study and the role of various parameters a Recent results As recent results we should mention Brabec's cell based on PPV and fullerene derivatives : at Siemens , he claims an efficiency around 5% [Bra 04] . Conversely to this wet route , dry deposition with small molecules can also be realised; so, with the structure ITO jPEDOTjCuPc - C60 jCa+AI, we obtained as photovoltaic parameters (Figure XI-20) : Vco = 0.45 V, Jcc = 0.768 rnA cm- 2 , II ~ 1.8% with ITOIPEDOTI3"(15nm CuPc+25nm C60)lAJ

, . . . . - - - - - - - - - -O.OOt-r-- - - - - - - - - - - - - - - ,

..E

1 - - - - - - - - - - -O,OOOS+------ - - - - - - - - "',·- ---1

u

-c g.lJ,6

I c ~

f - - - - - - -- - -

08

0

-0.4

-O,OOOS

1

-"'::--------- - - - - -1

17

:::>

U

'-------------o,.OO t 5~-----------------J

Bias Voltage

Figure XI-20. I(V) characteristic ofITO/PEDOT /CuPc - C60/Ca+AI.

338

Optoelectronics of molecules and polymers

4 mW cm- 2 as luminous flux. As we can see on the I(V) characteristic, an important serie s resistance drastically disturb the measurements; an electron transport layer at the cathode interface could improve the result s. With pure CuPc as an acceptor layer and a BCP layer acting as a blocking layer in relation to exciton diffusion at the cathode interface, J.M . Nunzi [Tro 03] obtained very encouraging results with an interpenetrating network made ofITO/ PEDOT:PSS/ CUPC:C60 (1:1) (40nm)/BCP (8nm)/Al (60nm) with annealing of CUPC:C60 at ISO °C under argon atmosphere for 20 min. Under irradiation of 100 mW cm - 2 , the optoelectronic parameters are as following : Voc = 0.50 V, Jsc = 11.5 rnA cm- 2 , FF = 0.37 and 11 = 2.14 %.

b Improving the photovoltaic properties of materials It could go via the implication of several different technological route s. Here we can cite five examples:

•

•

•

Adapting the band gap ofmaterials to solar light this step is possible using the various , available , synthetic routes to n-conjugated polymers with differing band gap s [Ronc 98]; Orientation of chromophores to generate an internal field An alternative method to effect photodissociation throughout the whole volume of a film was proposed by C. Sentein et al. [Sen 97] . The proposed route implicates the creation of an internal field due to the orientation of chromophore molecules by the continuous application of an external field, in a technique otherwise known as 'poling' . This provoke s in the photovoltaic device an effect of rectification which would stimulate photodissociation. This effect has been shown to improve a mono-layer device, based on non-optimised polythiophene used initially for photovoltaic conversion, by a factor of 40 [Sic 00]; Improving depo sition techniques to increase absorption capacities {Fie 00 J The octamer octathiophene (a -8T) , shown in Figure XI-21, was examined by C. Videlot and D. Fichou . They found that its properties varied greatly with respect to its orientation once deposited on a substrate. A pn-junction, in which the a8-T acted as the p-type semiconductor and a derivative of perylene was used as the n-type semiconductor, was used in the system LTO/a-8T/perylene derivative /Al . Between 400 and 600 nm, a photo-current (Ipe) was generated when the system was illuminated from the ITO side with polarised light. It was found that the current increased by a factor of ten when the a8-T molecules went from being perpendicularly aligned to being aligned in parallel with the substrate. Thi s effect was attributed not only to an improvement in the transport properties of the a8-T,

Figure XI·21. Representation of Cl-8T.

XI Organic photovoltaic devices

•

•

339

but also to a better coupling between the molecules and the incident light, an effect which improved their optical absorption ; Improving charge transport Following the disassociation of photogenerated charges, the problem of their transport to the electrodes, as we have seen, needs to be tackled. Given the low mobility of charges in organic solids, possible solutions may exist in the .use of ordered phases based, for example, on liquid crystals . Furthering this idea, the mobility of electrons being again low in polymers, an eventual solution may rest in the use of mixed organic/inorganic systems , in which the transport of the electron may be assured through the inorganic phase . Doping effect of n or p layers is also studied, as well with molecular materials (dyes) as with the ion beam implantation technique we have previously developed [Mol 98] to realize organic devices (diodes or transistors); Optimising the morphology of interpenetrating systems The optimisation of photogenerated charge separation requires, on the scale of exciton diffusion lengths, a perfect homogeneity of the donor-acceptor system . A solvent free method, using the co-evaporation of p-type oligomers and n-type molecules could reasonably result is such a structure ; the vacuum quality is a critical parameter really. It has also been remarked that the use of composites which exhibit concentration gradients , with donors and acceptors on opposite sides of a structure , should improve the collection of positive and negative charges at the respective electrodes.

6 Conclusion At the present moment there is a considerable push in research into the field of organic photovoltaics and it is with some excitation that more results are expected in the near future . On bringing together the various possible techniques, which are not necessarily exclusive to one another, the short term aim is realise structures which display efficiencies of the order of 5%-6% and have lifetimes of around 5000 hours . Activities based on organic photovoltaics should be possible on an industrial scale. And given the recently published results, researchers could well be there in the next few years.

XII The origin of non-linear optical properties of n-conjugated materials and electro-optical modulators

I Introduction: basic equations for electro-optical effects 1 Context Whilst this book is mostly concerned with coupling electron transport propertie s with emis sion and optical absorption propertie s, this Chapter deals specifically with the effect of an electric field (E), whether static or variable, on the optica l properties of an organic material. In particular we will look at the effect of electrica l fields on the indices of materials, and the changes in the intensity or phase of an optica l wave propagated through a polarised organic medium. The assemb ly of polymer based electro-optical modulators is then introduced. First of all though, as a general overview, we shall detai l the generally observed laws.

2 Basic equations used in non-linear optics [Sal 91] a Standard equations In general terms, a materia l subject to a static or low frequency electric field undergoes certain deformat ions. These typically result from either ionic movements in mineral b ased composites or from electron displacements in organic media (following electronic and molecular orbital deformations). Under the effect of an external electric field, as shown in Figure XII- I, the change in polarisation induces a modification of the optical indice of the materia l at a given frequency. There are two main types of effect that can be observed: • •

when the indice varies proportionally to the strength of the applied field the electro-optical effect is called the Pockels effect; or when the indice varies with the square of the applied field then the effect is called the Kerr electro-optical effect.

342

Optoelectronics of molecules and polymers

-B:fI e----

O~_fiC_ld

Electric field Figure XII-I. Coupling an electric field with an optical field traversing an electro-optical material.

On following the hypothesis that the variation in the indice is small with respect to the applied field, we can use a Taylor type development around values of the indice for when E = O. Using the notation thus: neE = 0) = nCO) = n,

a]

dn) = dE E=O '

gives :

I 2 neE) = n + a] E + -a2E + ... 2

(I)

For the reasons of notation which will become apparent in Section b: r = _ 2a] n3 a2 s = - -3 n

I

are the electro-optical coefficients which are such that: 1 3 1 3 2 neE) = n - -rn E - -sn E + ...

2

2

(2)

Typically, the 2nd and 3rd order terms are small, by several orders, with respect to n.

b The form of electrical impermeability The term electrical impermeability is particularly useful when describing anisotropic media (for example, following ellipsoidal indices). It is defined by 11 = llE r = I/n 2 . As on one side 11 = ~, that is ~~ = ~;, and on the other there is

1 2

~n=n(E)-n=--m

3

1 2

3 2

E--sn E,

we can write ~11 = ( d11 dn ) ~n =

(2 Isn32 ) = rE + sE2. E- 2 E + ... - n3 ) (-21m 3

XII The origin of non-linear optical properties

With

~11

= l1(E) -

11, l1(E)

= 11 + rE + sE 2 + ...

343

(3)

(The coefficients rand s are defined beforehand to arrive at this simple relationship for ~ 11, which introduces the simple coefficients r or s in front of E or E 2).

c Pockels effect and order of size In many materials the term relating to the 2nd order Kerr effect is small with respect to that of the I st order Pockcls effect. In effect:

I 3 neE) ~ n - 2"rn E.

(4)

Typically, r ~ 10- 12 to 10- 10 m y -I , that is from I to 100 pm y-I . If E 106 Y m- I, then the term 1rn3E is of the order of 10- 6 to 10- 4, a very small variation in the indice with respect to the effect E (the usual minerals considered are KH2P04, LiNb03, LiTa03 and CdTe) .

=

II The principle of phase modulators and organic materials 1 Phase modulator Phase modulation means that information, based on either yet) or E(t) , can be transferred using an optical signal by modifying the phase of this optical wave. An alternative method is with an amplitude modulator which varies the intensity of the optical wave by using parallel monochromatic luminous beams, i.e. a laser beam. When an optical beam traverses a Pockels cell of a certain length (L) subject to an electric field (E), as detailed in Figure XII-2, the wave undergoes a shift in phase: 2n

Ao

n(E)L,

(5)

where AO is the wavelength of the light in a vacuum . The introduction of eqn (5) into eqn (4) yields:
Optical field

2n = -nL Ao

2n I 3 -L-rn E Ao 2

=
rn 3EL n--, Ao

(6)

-; d (a)

Figure XII-2. Pockels cell under : (a) transversal (E (E = V I L ).

= V Id ); or (b) longitudinal polarisation

344

Optoelectronics of moleculesand polymers

with
rrrn3 VrrL
(7)

Accordingly, we can deduce eqn (8): }..o d V rr = rn3 L'

(8)

For a longitudinal Pockels cell, as in Figure XII-2-b, the field is now of the form E = V /L and the half-wave voltage, which is commonly denoted by Vkrr' becomes Vkrr = ~~. We can now use eqn (6) again in the form

V V . . - (longitudinal cell) .

(6')

Thus the shift in phase changes linearly with the applied tension. This we can use to modulate the wave traversing the cell. Generally, the value of V~, rt can be of the order of I to several kV for longitudinal modulators. Transversal modulators permit a lowering of the value of V rr through variation of the ratio ~ (d can be several microns and L can be around I em), so that v; ~ I to 100Y.

Comment If the applied field (E = E(t)) varies greatly during the transit time (T) of the optical wave, then obviously the optical wave will be subject to a varying value ofE during its passage through the modulator. The field E(t) which can be used is the

o:+----~---

V

Figure XII-3. Variation in dephasing(
XII The origin of non-linear optical properties

345

Figure XII-4. Positioning of electrodes with respect to the transmission in a transverse modulator. average field Eduring the two instant t and t + T. The size of the modulation band is generally taken as being equal to l iT. In order to limit this problem, the tension (V) can be applied at the side of the entrance of the optical guide so as act directly on the transmission axis . This is schematised in Figure XII-4. Modulation up to several GHz is possible using this technique.

2 The advantages of organic materials Other than the advantage that organic materials have in being easily shaped, they do present other advantages specific to this field :

a Advantages over inorganic materials As the inorganic materials used are normally ionic in nature, the displacement of these charges is slower relative to the movement of electrons alone, especially electrons delocalised in organic media. Ionic polarisation is relatively slow especially when compared to the response times of organic materials, and in particular rr-conjugated polymers, which can be extremely fast (of the order of femtoseconds) .

b Advantages due to the low dielectric permittivity of organic materials (er R:j 2) Figure XII-5 schematises the circuit used to control a Pockels type cell. The generator provides an excitation at low frequencies which is applied to the cell and acts, no less, as a condenser. Its losses are shown in the scheme by the parallel conductance G p . The RC , is small if C is small. This is favoured in organic materials as response time, t their permittivity (E r ) is also relatively small (s, ~ 2) . In addition, we have:

=

Vs Ve fc

I

=-------~

1 + Z(G p + jwC)

I

.

1+ JWZC

.

(With

Gp

«

wC).

The modulator acts as a low frequency filter, with a cut-off frequency (fc ) at (denoting S = [I x L] as the available surface of

= 21t~C" As C = £o~rS = £o£J1L

346

Optoelectronics of molecules and polymers

Figure XII-5. Schematisation of a Pockels cell control unit.

capacitance, see also Figure XII-2), we can see straight away that with organic materials that have a low value of Er , C is also small given that the cut-off frequency is high. Comment On introducing VIt, which insures the phase shift by rt and is such that d ' . !£!::!l. _ n3r _ " . V It -- l!l.. 3 -L' we amve at. v - -Zl 2 - constant, lor a given system. nr

n

It

~~

For any given system in which fc increase, VIt also increases (along with consummation). And the ratio fcAO/V It defines the figure of merit of the modulator.

c The state of the art Y. Shi at the TACAN Corporation, along with his colleagues at the Universities of Southern California (Department of Chemistry and the Centre of Photonic Technologies) and Washington, has shown that by controlling the form of the organic chromophores it is possible to obtain electro-optical modulators having half-wave tensions lower than 1V (V It ~ 0.8 V with VltL ~ 2.2 V em) [Shi 00]. The results also showed a large increase in gain (inversely proportional to Vi). And the considerable decrease in parasite signals allowed the operation frequency to be increased to 1000Hz, equivalent to a transfer of data of 100 Obits s-I . The decrease in VIt was realised by increasing the concentration of chromophores while simultaneously limiting electrostatic interactions (which reduce the electro-optical effect). This was made possible by inserting chromophore molecules containing bulky groups that contributed to steric effects. Other materials, notably those with octapolar charge distributions, have been used in attaining 'giant' non-linear effects [Med 95].

3 Examples of organic donor-acceptor non-linear optical systems Two principal effects are responsible for electronic displacements in molecules [Mer 95]: •

•

the induction effect which gives rise to a preferential attraction for electrons shared by atoms having a-bonds (in a range of electronegativities range based on the capacity of atoms to attract electrons); and a mesomeric effect which brings into play n-electrons or doublets of unshared electrons situated in p-orbitals. These charges can move from one bond to another and are not localised to the extent of electrons in a-bonds.

XII The originof non-linear optical properties

347

a The induction effect Briefly put, when two atoms are bonded by a a-bond, the more electronegative atom tends to attract the electron s in the bond towards itself giving rise to a dipolar moment. In addition, the polari sation effect can be transmitted along a chain of covalent in an effect which dimini shes with distance (and typicall y disappears beyond the third bond). The inducti ve effect is represented by the symbol + -+ with the arrow directed towards the negative end of the dipole, and in the opposite direction to the sign - ----+ +

: Sil·

b The mesomeric effect The structure of a molecule, in reality, is in between the two limited formulas which are both unable to describe the partition of electrons, given that n-electrons are more mobile than a-electrons.

Example: the isolated double bond

The carbonyl group is normally represented by:

""

..o / ;C= ..

Neverthele ss, 0 is much more electronegative than C, and can attract electron s more powerfull y. The following repre sentat ion is therefore also plausible:

""+ C -0: /

The rt-electrons are delocalised towards 0 and, in effect, this situation can only be written using both of the proposed forms. These two forms are called mesomer s and are both intermediate with respect to reality.

c Competition between induction and mesomeric effects and the example of substitution on benzene rings ct Influence of an electron donor: - NHz. resonating with the benzene ring chemic al representation schematises specific effects:

The

(i) a weak induction effect [N (Se" ) is more electronegative than C]; (ii) a strong mesomeric effect through the conjugation of the N free pair of electron s with the benzene ring; and (iii) the pair of electrons on N(:) tend to delocalise towards the ring.

348 Optoelectronics of molecules and polymers Result The electron cloud is delocalised towards the cycle and, in effect, the -N(:)Hz group acts as a donor.

....

(i)

Minor inductive effect (ii)

+

(iii)

.....

+

+

6·--6~"6 Strong mesomeric effect

~ The influence ofan electron attracting group : N02 In the case of N02 there is no free electron pair ready to delocalise towards the benzene ring. With 0 being more negative than N, 0 tries to saturate itself with electrons moving them from the ring. In effect, NOz is an attracting (or acceptor) group.

Induction effect

Mesomer effect

4 General structure of molecules used in non-linear optics [opt 93] As will become apparent when looking at the materials in more real terms, organic molecules which allow intermolecular charge transfer can give rise to a particularly high hyperpolarisability. These molecules are termed 'push-pull' molecules. They consist of donor and acceptor groups linked by a conjugated system, otherwise termed a 'spacer' (S) or a transmitter, as shown in Figure XII-6.

XII The origin of non-linearoptical properties

349

Dfonjugated systemL A (8)

--T

Figure XII-6. Donor-spacer-acceptor system.

III The molecular optical diode Here we will show that molecular optical diodes can in fact give a rectifying effect. Light polarised in one direction can pass, while light polarised in the opposite direction is 'blocked' . An analogue is, of course, a rectifying pn-junction, which rectifies by exploiting effects due to a passing current and the excitation developed by an electric field. In molecular optics, the created electronic response is FE = ndji ( = ~~) in which n is the density of electrical charges and dil = qdl is the dipole formed on displacing by dl an electric charge from its centre of gravity due to the passage of an optical wave. Comment In linear optics, the polarisation is tied to the optical indice by the relationship F = EO(E~ - I)E, with E~ = n2 for non-absorbing media. In absorbing media, = n*2.

E;

1 The centrosymmetric molecule The centrosymmetric molecule contains electrons which are highly polarisable. A good example is that of delocalised n-electrons, as found in benzene (C6H6). dji = qdl represents the dipole moment.

-q

dl

------+)

+q

Additionally, in one dimension, moving a charge (q) by a distance dl superimposes on the system a dipole moment of dil = qdl, as detailed in Appendix A-9, Section III. Similarly, moving a charge -e by -dl is the same as applying a dipole moment dii = edl , as shown in Figure XII-7. In addition, if we move electrons under an Initial electron position

.

Superpositioning a dipole d~ = ed!

Result: displacement of an electron by - ill=- 0'0

ill =0 '0

-8

0/

x

df=e ai 0';-';0 -8 +8

. .

o~ o

-8

-E

+

Displacement can be due to field (E) Figure XII-7. The equivalence of chargedisplacement, polarisationgenerationand vice-versa .

350

Optoelectronics of molecules and polymers

Figure XII-So Representation of the electronclouds in an unpolarised benzene molecule.

electric field (E), with direction -dl, that implies the application of dji = ed! and thus a e,olarisation P= ndji is generated in the opposite direction, depending on dl. If E = 0, the benzene molecule electron cloud has the form described in Figure XII-S o We can now consider the same benzene molecule under an electric field (E), as in Figure XII-9. _ With respect to the axis of time, successive lobe s due to the function P remain symmetric. There is no 'rectifying' effect observed at this size of optical field (E), whether it_be a weak field in which case P is directly proportional to E as P = EO(E r - I)E , or whether it be a strong field in which case the polari sation stays as a unpaired function (with successive + and - lobes) an9 a non-linear effect is introduced simply by the unpaired harmonics at the level of P.

2 Non-centrosymmetric molecules Now we can look at the same benzene molecule which has, however, been modified by the addition of electron donor (D) and acceptor (A) groups so that it has become non-centrosyrnmetric. This modification results in the displacement of the electron cloud towards the attracting, acceptor group. Any effect exerted by an optical field will be exacerbated by orientating the optical field so that it displaces electrons toward s A. Inversely, its effect would be minimis ed by orientating the field so that it tends to move electrons towards D. Figure XII-IO schematises these two, opposing effects . As detailed in Figure XII-II , Fourier analysis of the resulting polarisation wave shows that it contains, that for non-centrosymmetric molecules, a fundamental, a second harmonic and a c~nti n_uous component. In a one-dimensional medium and when the optical wave is E = EW , the term s for the polari sation wave should thus be written in the form :

wherein Xa is the absolute dielectric susceptibility and such that Xa = EO Xr. Here Xr = (Er - 1) is the relat ive susceptibility, hereon denoted simply by X.

XII The origin of non-linear optical properties

351

E strong field

~l-f'--

-- -

....

---""I.--

---7I',--

-- - --.

t

:" -- ---j : I I

first alternation

I

/

I

I

Weak field

I

second alternation

- I

~1

ill --+tdi df 1_

--+

p=

--+ ndf

polarisationunder strong field

e \£) -- -- -

_- - -

polarisationunder weak field

t

.

Figure XII-9. Polarisation by strong and weak fields of a centrosyrnmetric molecule .

3 Conclusion The second order non-linear polarisation (p(2)) is proportional to the square of E (the term E 2 introduces cos 2 cot = 1+C~S2wt) . It is only with a non-centrosymmetric molecule that that either X2 or Xa2 can be zero . In addition to the phenomenological evidence described above, we can also perform a direct demonstration of this effect [yar 89]. _ _ If a molecule is symmetric, then changing from E to -E is effectively the same as moving charges in an exactly similar but symmetrically direction, so:

(I)

352

Optoelectronics of molecules and polymers

fiR E

polarised wave

r '-~ PN !r I~

I

first alternation

exaggerated electron disolacement

!

I

second alternation

I

small electron movement

Figure XII-tO. Polarisation of a non-centrosymmetric molecule .

I

ij Fundamental

Second harmonic

Continuous component

Figure XII-H. Resolution of polarisation wave for a non-centrosymmetric molecule.

XII The origin of non-linear optical properties

353

Additionally, the definition of p(2) gives: }:::} p(2)(E) = p (2) (_ E) p(2)(E) = Xa2E.E P(2)(-E) = Xa2(-E)-(-E)

(2)

And finally: from eqn (1) : Xa2E.E = -Xa2(-E)-(-E)} . ~ E ~ (E~) (E~) :::} Xa2 = -Xa2, that IS Xa2=O. . and from eqn (2): Xa2 E . - Xa2 -

IV Phenomenological study of the Pockels effect in donor-spacer-acceptor systems Here we shall describe a molecular system containing n-electrons and establish , directly, the relationship between modifications of the function indice with respect to the applied field, otherwi se known as the Pockels effect. The establishment of the same type of equation, using more conventional methods , is detailed in Appendix A-II . We will use an optical wave (E W ) which propagates in the Oz direction and is polarised in the Ox direction . Figure XII-12 indicates how the electrodes are placed about the active medium .

1 Basic configuration a Molecular system with prior orientation of It-orbitals in the direction Ox A static polarisation field (Es) is applied using 'poling' , that is to say the field is applied during a descent in temperature of an electro-optical material from its glass transition temperature (Tg) to ambient temperature (T) to give the result shown in Figure XII-l3-a.

b For systems also containing a D-A pair Here we are considering the system surrounded by the dashed line in Figure XII-l3. The electronic cloud is deformed and the centre of gravity for the charges is moved

·z

Figure XII-12. Configuration of an active medium placed between electrodes parallel to the plane yOz and traversed by an optical field polari sed with respect to the Ox direction.

354

Optoelectronics of molecules and polymers

r - - - - - - - - -- - - - -

(a)

I

I D

I I I

I I

I

I --=--:-:r;---

Y

Y Es

I +

A Symmetri cal molecular system. (Centres of atom ic and charge gravitie coincide ).

L

x

x (h) and (c)

(d)

I I

I I

I I

J

D-A system

Figure XII-B. The position of the centre of gravity of electronic charges (GE) for: (a) a symmetrical molecule; (b) a non-centrosyrnmetrical molecule whetheror not (c) in thepresence of an opticalfielddirectedtowards Ox; and (d) the same as (c) exceptingthat the fieldis directed towards - Ox. by x > 0 so that OG E = x. The greater the donor-acceptor character of the additional groups, and in effect the greater the size of the electron cloud, the higher the value reached by x (Figure XII-13-b).

c In the presence of an opticalfield The ions which are attached to atom s within the solid materi al and in effect it is only the electronic cloud which moves, in a manner similar to that shown in Figure XII-13-b , once the sys tem is under the influence of an optical field. The movement of the electron cloud is with respect to its centre of grav ity, already moved by the O-A pair. With GE displaced under the effect of both the O-A pair and the optical field by a distance x we thu s have fr = -kx (Figure XII-I3-c, identical to Figure XII-I3-b).

d The effect of a static electric field (Es) applied in the direction - Ox Here we consider the situation pre sented in Figure XII-I 3-d. The electric field goes against the return force fr and has a tendency to displace GE towards x > O. Thi s displacement occurs as strongly as the charged electron cloud is large due to coupling strength. In effect, as we have seen in Sub- sect ion b ju st above , the value of x tend s to be high. We can therefore write that the force generated by E, is in the form fES = +Ox ; the force fES increases prop ortionally with Es .

e Result of combined forces The result of the force s (fR) with respect to Ox of force s fr and fES is thus in the form fR = fr + fES = -kx + Ox.

XII The origin of non-linear optical properties

355

With k = mW5 and D = m&2 (which is such that &2 ex: Es , as D ex: Es): (1)

fR = -m(w6 - &2)x .

Comment J If the resulting displacement x is, in effect, as large as Es and thus also D(D ex: x), the term Dx = &2x varies similarly to x 2, that is as an inharmonic term. Comment 2 With an optical wave polarised in the direction Oy, the fundamental relationship for the dynamics of the system can also be written for the direction Oy. Here the field E, is put into terms of Oy and this effect may be greater than that in the direction Ox given the anisotropy of molecular orbitals (see Comment 2 at the end of this Chapter, Section IV-4).

2 Fundamental equation for a dynamic system Here we will assume that in the permanent regime under study frictional forces (f t = -mr*.) are negligible due to r « [co - wol (the further from the resonance point we go (co - wo) the more r tends to zero). Therefore, the fundamental dynamic relationship for an electron with respect to Ox is:

(2) W

Only the qE component contributes to the Coulombic force . The effect due to the static field (E s ) was taken into account within the term m&2x, which is included as part of the return force (fr ) as the electronic effect of E, was considered as a modification, or rather perturbation, of the return force . With E W = Eo cos cot, the solution under a forced regime x = Xo cos wt( = xoeiwt that is = -w2x) gives us:

x

w

2 2 2 eE ([wo-& ]-w ) x = - m

3 Expressions for polarisability and susceptibility With N as the electron density, a polarisation of P caused by the displacement of electrons by a distance x along Ox can be determined using P = -Ne x Ux and 2 w2 = Ne in p

com

(3) Typically, we set

w2 X.w =

2

P 2'

wo-w

(4)

This is the term which appears in a linear polarisation of a system not subj ect to an additional static polarisation (E s ) ' In this case , we have fES = 0 and the term in 1)2

356

Optoelectronics of molecules and polymers

disappears . The eventual expressions of the polarisation (p(w)) is thus in the form: -(w)

p

2

wp -w -w = £0 wo-w 2 2 E = £OXw E .

On assuming that 8 is small with respect to [co - wol, placing it outside of the resonance zones. And on using eqn (3) then with eqn (4): (5)

4 Expression for the indice-and the insertion of the electro-optical coefficient r With £r = n~ = I presence of Es :

+ X, we have as a relationship for

On introducing nw = .)1

the indice in Ox and in the

+ Xw, the indice in Ox and without E, gives (6)

Finally: (7) 2

· h Xw = ~ h WIt 2 ' or rat er 2 wo-w

As nw = .)1

1

-2-wO- w2

2 2

Xw, we h ave un A xwS- . = = nx - n., = -

+ Xw {} Xw = n~ -

w~

2nww~

1, we can write

(8) As indicated above, 82 ex: Es = aEs, so that to a first approximation-neglecting I in front of n~ (given that with the presence of chromophores the medium is, intrinsically,

XII The originof non-linear optical properties

357

highly polar so that its permittivity (s, = n~ ) can be relatively high by even 3 or 4 orders)-we have: n4

~n ~ ~a 2nw wp

~n in which r

== rp ~

Es ,

in which ~n is in the form

I

= 2 m3Es

~ Wp

(9)

and is the electro-optical coefficient, and n.,

== n by notation,

an indice following the direction of the polarisation of the optical wave and measured in the absence of the polari sing field Es . ~n is proport ional to Es , and eqn (9) is identical to eqn (4) and well characterises the Pockels effect. Comm ent 1

as d

There is another, possible way to represent the Pockel s effect:

(:2) = d(x - 2)= -2x- 3dx, we can write

~ (:2) =Id (n12)1=2n- 3~n =2 ~~

and on using eqn (9) leaves

~ (:2) = rEs ·

(10)

Here we recover the classical expression used to define the electro-optical effect, which was evolved from the ellipsoidal deformation of indices at a low frequency electric field (linear variation within the electric field of coefficients

[n'r 1from an

ellipsoid of indices). Comm ent 2 Variations in the indice due to the application of E, can themselves be variable. This depends on the orientation of the static field E, with respect to the direction of polarisation of the incident optical wave and anisotropic effects can appear. However, in reality it can be advantageous to polarise the wave in the direction Oy perpend icular to E, (which follows Ox). The deformation of orbitals by Es , only for the propagation of the same wave but polari sed in the direction Ox, can have a greater effect on propagation along Oz than along Oy. In the adjacent scheme, we can see that the p-orbital retains the same elongated form in Ox (admittedly slightly longer in the presence of Es ) , but is also much 'skinnier' in the presence of Es .

~ lx

.,0

358

Optoelectronics of molecules and polymers

V Organic electro-optical modulators and their basic design 1 The principal types of electro-optical modulators An electro-optical material can be used as a polarised wave phase or amplitude modulator depending on the design geometry chosen.

a Phase modulator The phase modulator demands the more simple structure . It consists of a modulator arm constructed from a simple optical guide, onto which the incident wave falls. The modulation is performed by applying a transversal tension, which induces a phase delay (~
b Amplitude modulator Phase shifting alone does not change the intensity of the optical wave subject to modification. However, here, the modification of the intensity of an optical wave is possible by using, for example, a Mach-Zehnder amplitude modulator. This device can be inserted into an arm of an interferometer. As shown in Figure XII-IS, the amplitude of the incident wave (L) is equally shared into two waves, each with intensity! Ii, down two arms neither of which have been subject to an electrical field E. The from of the modification induced by E on

Exiting li&ll' mod ulated by E

Figure XII-14. Schematisation of a phase modulator.

XII The origin of non-linear optical properties 359

7

electrodes =>cos 6'1'

Figure XII· IS. Schematisation of a Mach-Zehnder type amplitudemodulator.

1

one of the arms is Ii cos Ll

1 1

J"

2 cos 2 x ~"21 ( 1 + 1 - 4X ) = I - x 2,

2

Ll
and

x

= 2'

we can obtain: 10 = Ii [1or in other terms, ~

=1-

Ll:

2 ]

= t, [1- 4~; y 2] = Ii -

Kliy2,

Ky2.

If Y = Yo cos cot, then y2 ex: cos 2 wt ex: cos 2wt , and thus ~ ex: C I The intensity 10 is thus modulated by a frequency 2w.

-

Cz cos 2wt.

2 Figures of merit As already described in Section 11-2-c, the half-wave tension, in the form Y Jt = d Ao ~ -L for a transversal polarising modulator, should be as low as possible (in rn principle around 1V). Y Jt is often used to characterise the performance of a modulator. Given the expression for Y rt - in order to decrease this parameter it is in our interest to decrease the coefficient ~ and in effect increase within reason the length of the modulator arms. d is normally around several microns and L ~ 3 em . In addition, the factor m 3 needs to be as large as possible to decrease Y Jt, and can thus also be taken as a figure of merit as in FM = m ', and is expressed in pm y -I . Figure XII-16 shows the figure of merit as a function of the wavelength of the optical field for several different materials (from [Bos 95]). Figure XII-I? shows the general configuration of optimised stilbene and benzene in addition to values of rand n at given wavelengths of light. D and A, respectively,

360

Optoelectronics of molecules and polymers

Figu re of

l1le~

-------

10'

KnhO.1

Optimised st ilhene 101 Optimised benzene

"-----------=--, avele ngt h (nm)

500 Figure XII-I6. Figure of merit MF

= n3r (pm V-I) as a function of 'f. (nm).

D-O--A Optimised benzene r (501 run) n (501 om)

=210 pm yol =2.2

Optimised stilbene r (579 nm) 1150 pm V·I n (579 nm) 2.4

= =

Figure XII-I7. Configuration of optimised benzene and stilbene with example values for r and n.

Acceptor A CH3 CN

OCH 3

Donor D NH z N(CH)3

12

20

56

60

35

67

197

218

Figure XII-IS. Values of ~ (10- 4 m V-I) for D-C6H6-A with varying donors and acceptors

A ('f. = 1064 nm) (from [bos 95)).

XII The originof non-linear optical properties

361

donor and acceptor group s exhibit efficiencie s in the following orders: D, N(CHh >

NHz > OCH 3 > OH ; and A, NO > NOz > CHO > CN. The second order molecular polari sabilitie s (B), expressed in 10- 4 m V-I , are given in Figure XII-18 for various para-substituted benzenes. ~, which is also called first order hyperpolarisability, is defined by the relati onship:

p = 110 + Eo([a jEf + [ ~ j Ef Ef

+ [vjEfEf Ef + ...)

in which 110 is the dipole moment of the fundamental state, without deformations and therefore without contributions from the various first [a], second [~j , third [vj and so on orders of polaris abilitie s. raj, [~j and [vj are tensors in effect and Ef is the field localised at each molecule. Macro scopic polari sation follow s the form (see also Section IlI-2 ):

Once molecules are organised in systems with centre s of inversion, resulting in centro symmetric dispositions, strong, microscopic non-linear effects (high value of ~) become ineffective at the macroscopic level (XaZ is small).

3 The various organic systems available for use in electro-optical modulators a Benefits and losses with organic media As already mentioned in Section II, polymers exhibit several attractive qualities with respect to their use in electro-optical devices such as their low cost, their ease of synthesis and manipulation (using spin-coating techniqu es for example) in addition to their interesting physical characteristics, which include their low functioning voltage and dielectri c permitti vity (Er or the order of 2 to 4), the latter of which makes it possible to envisage a good adaptation to the respon se times between the electrical excitation and the optical wave. Certain inconveniences found with these materials should, however, not be ignored . They include the elimin ation of the isotropic character in organic (amorphous) media, most notably in polymers (rt-conjugated included) which tend to form centro-symmetrical systems in which non-linear second order effects also tend toward s zero. In addition, in order to activate the non-l inear properties, ' poling' microscopic aggregates with an electric field is necessary. Any alignment of such aggreg ates breaks down the centro- symm etry. The material does, however, retain a certain instabil ity due to the ability of thermal effects to agitate and disorientate any aligned aggregates. The stabilisation of ce ntro-sy mmetry is the main cause of problems at an industrial level, especially when the material s otherwise exhibit high chemi cal resistance, and a reasonably low sensitivity to humidity and to oxidation of n- bonds. For the present moment , one of these materials used in organic modulators is polyimide [Gar 99j.

362

Optoelectronics of molecules and polymers

b Thedifferent systems Most crystallineorganicmaterialsare centro-symmetric and thus uselesswithrespect to second order effects. Non-linear moieties are thus often used incorporated into an amorphous polymer, which is vitrified directly following preparation into a film. The electro-optical performances are conditioned by the concentration of the chromophores (non-linear D - n - A moieties) used which can be inserted into the final material. Different systems, using this technique, can be prepared: •

• •

'guest-host' systems, as shown in Figure XII-19-a, are prepared from a mixture of an optically active medium mixed with a polymer (which necessitates solvents such as toluene or chloroform). Solubility parameters control the final chromophore concentration; systems prepared by grafting chromophores onto the back bone of a polymer, as detailed in Figure XII-19-b; and covalent systems obtained via the formation of covalent bonds between chromophores and polymerschains, as shown in Figure XII-19-c.

A stepcommonto the use of thesesystems is that once preparedthechromophores must be alignedso that the material loosesas muchas possibleits amorphous character and improves its electro-optical efficiency. The effect of this procedure is shown in Figure XII-20-a and -b for guest-host and polymer grafted systems. In order to

(a)

(b)

Figure XII-19. Different possible systems: (a) 'guest- host'; (b)chromophore grafted to back bone chain; and (c)chromophore inserted into polymer chain.

XII The origin of non-linearopticalproperties 0

363

0

;01

,

;01

>

w (a)

0 ;01

,

»

w

(b)

Figure XII·20. (a) 'Ouest-host' and (b) grafted systems, after chromophore orientation by 'poling' .

get around this stage, or to stabili se the orientation obtained by using this method, mesogens (polymer mesogens) can be used which orientate themselves in a given direction (in the liquid crystal state) . In theory though it seems that this latter method may not give satisfying electro-optical performances.

VI Techniques such as etching and polyimide polymer structural characteristics 1 Paired materials: polyimidelDRl As a material for a polymer matrix, polyimide presents numerous advantages including its low optical losses, low water permeability, low refraction indice (1.56 at 1.3 urn) and dielectric constant ( ~3 .2 to 3.5) and good thermal stability (T g = 290 "C). The method used for obtaining films has been well establi shed . In the case of polyimide 6FDA /ODA (otherw ise known as PI 2566 and supplied by DuPont) which is detailed in Figure XII-21 , an adherence promoter is used first of all (VM 651 promoter spread at 2500 rpm for 30 s), then an initial layer of PI 2566 is spread at 500 rpm for 5 s, and then finally the PI 2566 is spread at 200 rpm for 30 s. Following this step is an annealing step at 120 °C for 30 min on a heating plate and then a second heating stage at 300 °C in an oven for at least 1 h. Turning to one-dimens ional chromophores, a good example and representative molecule is Disperse Red One (generally called DR I) (Figure XII-22) , a well known

Figure XII-21. The monomer unit structure of polyimidc6FDA-ODA (PI 2566, DuPont).

364 Optoelectronics of molecules and polymers DRl:

Figure XII-22. The structure of DR!.

nitroa zobenzene based dye. It has been well characterised arid widely used for the last decade.

2 Devicedimensions-resorting to lithography a The dimensions a Intensity modulator Figure XII-23 shows the commonly used Mach-Zehnder configuration for an amplitude modulator. The sizes, shown in the Figure, have been determined with respect to both guiding conditions (mono-mode) and the need to produce a significant electro-optical effect. Assuring the length of the component (to obtain the required effect and reduce VIt), while retaining a guide depth of around 1 micron (to verify guiding conditions) complicates the experimental preparation of such a device . Lithographic masks are used which have to work well at very different scales: of the order of microelectronics and of the order of several centimetres. The technology used for motifs results in poor quality preparations. ~ Phase modulator The simplest device to prepare is that based on a single arm, as it consists of an optical guide sandwiched between two electrodes. In order to obtain a good electro-optical signal and to have appropriate guiding conditions, the dimensions of the device can resemble those shown above for the amplitude modulator. A simplified component can be obtained by using a cladding material to cover over the device made from a polymer which rests in contact with air. This type of guide can be particularly useful in determining losses due to the polymer (such as polyimide, polyimidejDRl etc.). The geometrical form of the guiding conditions were determined by Petermann [Pet 91 and Pet 96]. With the expression

I .S ~m 1\ - 250 ~m }rE~;;;~------------~---· j,{-- ---Wll\1!" -------- ----~

Figure XII-23. Dimensions of a Mach-Zehnder amplitude modulator.

E it wan

XII The origin of non-linear optical properties

365

the mono-mode guiding condition is tjh

w

R = - < 0.3 + ---;===~ h -

) 1 - (tjh) 2

Definitions of parameters for these equations are detailed in Figure XII-24 . For polyimide using the parameters detailed above, we have H = 1.49 (which holds well to the condition H 2: 1) and R = wj h = 1.25, a value below that derived = 1.44. from 0.3 + I- (t/h) The distribution of field lines, obtained equally well from calcul ations or experiment , is shown in Figure XII-25-a. What is particularly interesting is the way in which the lines are well confined within the rib guide. Figure XII-25-b shows the critical evolution of the ratio w j h with respect to h for different values of the coefficient r = tjh for a ribbon guide. A mono -modal regime is reached when (wj h) = 1 for r = 0.5. For polyimide, when r = 0.75 and (w jh) = 1.25 the mono -modal regime is attained.

k

3 Etching Once the dimen sions of the device s have been determined, they can be prepared by photolithography (see Chapter VIII, Section II-2) using masks of an appropriate size and quality. Given the research already gone into preparing etched surfaces exhibiting vertical walls, in principle it should be possible to use dry etching techniques to attain an excellent level of anisotropy. The speed at which the etching can be done depend s on the rate of pulverisation, which in itself depends on two component parts . One is physical and is based on the inert ions used such as argon, and the other component is chemical and is based on reactive reagents , for example oxygen ions used for polymer etching. Appendix A-9 detail s the principal mechanisms and characteristic parameters used . As described therein , the etching of polymers (polyimides) by ion beam s used for microelectroni cs must be performed using parameters close to those used in plasma etchin g (RIE etching). Figure XII-26 indicat es (for polyimide PI 2566 ) the rate of etching speeds with respectto the energ y (E) of the incident ion beam (3 keV :s E :s 6 keV). With a current density J = 0.4 rnA cm- 2, and a liqu id nitrogen trap set close to the sample so as to .-'" -_ ,, ,, , ,,

.

-_

-. ,,

,

"0=I: ail' " . = 1.56 : Polyimid e

": =1.45: SiD :

Figure XII-24. Parameters used in the Petermann conditions (mono-mode guide).

366

Optoelectronics of molecules and polymers

Iulti-rnodc region (a)

r=O.8

2

L...--_......,,

..:L.. . .

(b)

h

Figure XII-25. (a) Distribution of fi eld lines in a rib shaped guide; and (b) mono- and multimode regions.

cool the sample, limit its degradation, trap excess reagents and condense reaction products, a comparison of etching speeds has been performed for various ion beams (see also with Figure XII-16 in which the characteristics of the techniques used are shown), including: • • • •

dry etching using Ar + ions; Reactive Ion Beam Etching (RIBE) using 0 + ions without a trap; Ion Beam Assisted Etchin g (IBAE) using Ar " ion beam with a flux of molecular oxygen; and IBAE with a beam of 0 + ions and a flux of molecular oxyg en. As we can see in Figure XII-26:

• •

the differen ce between dry etching and RIBE etching is as great as the energ y E is small; and there is a step between the optimised energ y for RIBE etching (E :::::: 3 keY) and the energy used for the optimised IBAE process (E :::::: 6 keV). Thi s differen ce can be accounted for by the energy of 0+ ions which generate reactive species (0 ·) by colliding with the oxygen flux. The energ y transmitted throu gh colli sions

en

-

g>

E

"0 Q)

Q) c. ..... I

.s

:c .« ........ (.)

W

1400 1200 1000 800 600 400 200

a 2

4

6

Figure XII-26. Etching speeds compared using different techniques.

XII The origin of non-linear optical properties

367

between 0+ and 0 is of the order of 3 keY (taking into account the kinetic transfer coefficient y = 1/2), in precise equivalence with the energy of 0+ ions used to optimise 0+ RIBE etching . Finally, the optimised parameters obtained were : RIBE 0 + etching: E = 2.5 keY (with J = 0.4 rnA cm- 2 and realised using a similar method [Mou 00)); and IBAE 0+ etching: E = 6keY (with J = 0.4mAcm- 2 realised also for an ion beam current density) .

• •

4 Examples of polymer based structures [Cor 01] a Polyimide guide (as passive material) and characterisation of optical losses

11,,=

5~m

t um

6~m

I

280~~

I .....-------.. 6FDAIMPDA n, = 1,.575

t

4~m

L+":;"'~

' iO z n1 = 1.46

Si Ori entated

< 100>

" Figure XII-27. Geometrical structure of electron microscopic image of a polyimide guide.

c

.9

e.o

I

.g '"

,. ..et,

, L-

. .P. (a)

...J

"

11'I x-axis on guilt: (arbitrary unit)

,)Oz•

(b)

Figure XII·28. (a) Appearance of transversally diffused light; and (b) the corresponding absorption of the optical signal along the guide length.

368

Optoelectronics of molecules and polymers

P(db) =10 loglO(Po!Px)

· 12 dB {

19 18 17 16 15 14 13 12 11 10

PI 6FDA/MPDA

_.

-

-,

- - - ,...--

.,..,

-. -

,... .

-

-0

0.5

1

1.5

2

2.5 X (em)

Figure XII-29. Measuring propagation losses using the 'cut-back' method (LPropagation :::::: 2dB em-I).

b Modulator structure with polyimidejDRI as the electro-optical material

400nm

6pm

3pm

400nm

280pm

(u )

I •

Figure XII·30. (a) Scaled representation of a polyimidejDRI phase modulator; and (b) light injected into the same phase modulator equipped with electrodes.

VII Conclusion Using electro-optical materials such as lithium niobate (LiNb03) [Pap 85], the method by which modulators worked was well understood from the beginning of the 1980 s. However, the rapid development of active organic materials during the last decade or so indicates that completely organic modulators will be soon available. These hold the promise of being relatively cheap to fabricate and could be used in fibre optic networks either directly with the user or through a Hertzian process. Their exceptionally fast response times, originating from the electronic processes involved, will probably be required in response to the demand placed on communication systems, a demand

XII The origin of non-linearoptical properties

369

LED (a)

R emitter

(d) (b)

Rar e earth

hv radi tive

(c) Transfer from ligand singlet and triplet sta tes to Er (possibly 100 % ?)

Fundurnental level

Figure XII·31. Future prospects: (a) schematisation of an all-organic electro-optical device; (b) schematisationof an emitter; (c) principal of erbium emission; and (d) photolithography of polyimide which may permit network fabrication .

necessitating a doubling in optical telecommunication networks every 18 months in a process nicknamed ' Moore 's optical law' [Tou 01] . Figure XII -13 shows the possible structure of an all organic device [Umop 99]. The emitter could be fabricated using the layering shown in Figure XII-13-b and using the current knowledge detailed in Chapter X. The transfer of singlet and triplet states from the ligand q to erbium could be highly efficient, as detailed in Figure XII -13-c . The filter, within a network, could be fabricated using a focalised ion beam (FIB) machine,

370

Optoelectronics of molecules and polymers

giving 1.5 urn mono-mode guides with polyimide derivatives. Figure XII-13-d shows a photo of the derivative 6FDA-ODA as a pertinent example. The electronic servocontrol needs to be constructed taking into account the behaviour of the polymer based electro-optical modulator. In simple terms, experiments are devised in order to model derivatives and their characteristic behaviours and any secondary opposing reactions. In order to measure the dephasing attained at a given applied and continuous tension, a triangular and periodic signal can be directed at one arm of a Mach-Zehnder device . Then the modulated sinusoidal exit signal is recorded and once again another triangular tension is applied but this time with the continuous component. The recorded signal, again sinusoidal can be compared with the previous signal and the difference in the two accorded to the continuous component. We can finally say that the actual progress made recently will allow the fabrication of such devices without any major obstacles. As we have already mentioned, one problem does remain and that is the stability of organic materials. The continuous improvement in their qualities should, however, allow the emergence of devices very early in the 21st century.

Appendices

A-I Atomic and molecular orbitals

Introduction This appendix proposes a relatively simple construction of atomic and molecular orbitals for a covalently bound solid. And is aimed at readers relatively well versed in phy sical electronics. An example is made of the hybridi sation of carbon orbitals.

I Atomic and molecular orbitals 1 Atomic s- and p-orbitals Speaking approximately and with respect to atomic orbitals, we can imagine that each electron within an atomic move s in a spherically symmetrical symmetry. This is the result of a potential from the nucleus which varie s with I/r, represents the overall effect of all other electrons and is, to a first approximation , spherical. The electronic state is represented by the wave function (\IIn,l,m) which is dependent on the three quantum numbers n, nand m. The energy level depends only on n and I while the degree of degeneration is equal to the number of values taken on by m. When I = 0, the atomic orbitals are called s-orbitals and the wavefunctions vary only with n, as in \IIn,l,m = R(r)8 (t)) (m = Rn,080,oo, in which 80,0 = fando = ~ . These wavefunctions do not depend on e nor on <po Only R(r) takes on different values as the principal quantum number n varies . The s-orbital therefore take s on the spherical symmetry shown in Figure I .

=

=

When I 1, m -1 , 0, I and the orbitals are called p-orbitals: m = 0: 8 1,0 = (J6/2) cos e and <1>0 = 1/ J21t m = 1: 81 ,1 = (J3/2) sin e and <1>1 = (1 / JJ!) cos
-1 = (I /JJ!) sin
374

Optoelectronics of molecules andpolymers z

z

y

)'

x

x

Figure 1. s-Orbitals. If R(r) represents the function Rn , ] , as m varies, invariably the wavefunctions for the three preceding p-states are of the form: (I) : llIo =

~2'1II-; R(r) cos e

Iff .e '

(III) : Ill_I = -

2

Iff

(II) : Ill] = - -R(r)sinecos

-R(r) sm sm

If we set R(r) = R(rr) so that rr has a value defined by the relationship J~f IR(r)1 2r2dr = 95% (which indicates that the probability of an electron being

within a sphere of radius rr is equal to 95%), then we can say that the three equations represent the orbitals in a conventional manner.

Comment It is worth noting the placing of points M as defined by III = OM in Figure 2. As an example, we'll use the relatively simple relationship I

(3

---+

llIo ="2'1 "2R(r)cose = II OM II = OM. Here D is at the origin of the Oz axis, where cos e = I, and therefore OD = i[irrR(r) . Now OM = ODcose. If e E [0, n /2], OM > O. The M points are now placed with a varying e in a sphere above the plane (xOy). If e E [n /2, rr], cos e < 0, then the M points are in a sphere below (xOy). The signs of the orbitals llIo, Ill] and Ill_I can be obtained directly from the signs taken by the variables x, y, z.Todo this, we can use the relationshipbetween spherical

---7l:iI~-""' )'

p,·Orbital

Figure2. Shape of thepz-orbitaI. The + and - signs concern ljJo which follows cose.

A-I Atomic and molecular orbitals

375

z

m

Figure 3. Axes normally used for spherical co-ordinates.

co-ordinates and the Cartesian co-ordinates indicated in Figur e 3:

x . - = Sin 8 cos

. 8 . - = Sin Sin

z

- = co s 8 r

So for example, if cos 8 < 0, then z < 0 and Wo < O. For f er) =

~ R~r)

1ft

> 0 (with r :::::: rj ), the functions Wo. WI and W_I can thu s

be written in the following way:

= X = x f er) = wpx (Wpx > 0 or < 0 depending if x > 0 or x < 0) W_ I = Y = Y f er) = wpy (Wpy > 0 or < 0 dep ending if y > 0 or y < 0) Wo = Z = z f er) = wpz (Wpz > 0 or < 0 depending if z > 0 or z < 0)

WI

We can at last schematise in Figure -4 the s- and p-orbitals usin g r fixed to the value rr (whic h is such that f~r r 2 IR(r )1 2dr = 95 %): To co nclude, the s-orbitals are non-directional , unlike the p-orbitals.

Comment The orbitals we have shown so far concern only one ele ctron per orbital. They are the orbitals wh ich we will go on to use when con stru ctin g molecular orbitals from the sharing of one electron per atom in co valent bonding. However, if we were z

z

z

).

x

Orbital odal plane

x

ns

x

np, (y 0 z)

x

np),

np,

(x 0 z)

(x 0 y)

Figure 4. The ns and np orbitals. Note: the nodal plane is the plane in which there is zero probability of finding an electron.

376

Optoelectronics of molecules and polymers

to consider atomic orbitals which contained more than one electron, or i electrons, then the orbitals will take on the form \II = TI i\IIi in which \IIi are the wavefunctions for each electron. Prior to being anti-symmetric (Pauli), these \II functions are used in the Slater determinateform. 2 Molecular orbitals

Hi

a The example of the simplest molecular system

In the simplestscenariowe ignorethe possibleoverlapping betweenadjacentorbitals, which would otherwise have to be taken into account using the supplementary condition aE/ak = 0 in which energy is minimised following a variationmethod.With the two nuclei in our system placed-and fixed-at RI and R2 and the sharedelectron at r, as shown in Figure 5, we have the Hamiltonian for the system !i2 e2 H = --b. ~ 2m 41t£0(Jf - RII)

e2

41t£0(lr - R21)

If
Hi

• •

if the
---+

electron is localised on the nucleus (1), then
If the electronis localisedin betweenthe two nuclei, then the solutionhas to come from a mixtureof the two precedingstates. Wecan thereforelook for a wavefunction

Hi

HI
+ C2
= Elc,
+ C2
On multiplying the left of the equation by (
-e D

(2)

r

(1)

Figure 5. Layout of ions and the electron in

Hi.

A-I Atomic and molecularorbitals

377

On using the simplifying assumption of negligible overlapping, i.e. S = (
Similarly, on multiplying the left hand side of the same equation with r (
+ C2

°

(IV); and

(H22 - E) = 0.

(V)

In addition: •

we know that the wavefunctions
•

the problem (and thus also the Hamiltonian) remain invariant by permutation of the nuclei A (indice I) and B (indice 2), so that HII = H22 = - Ct.

As the system of the two eqns (IV) and (V) contain two unknowns, the solution is admittedly non-trivial as its determinant is zero. Using our notations, we obtain the equation (o + E)2 - ~2 = 0, which gives rise to two solutions for the energy (o + E) = ±~ that can be written using the form: EL = -Ct - ~ (the lowest energy corresponding to a bonding level as Ct and ~ are positive); and EA = -Ct + ~ (the highest and therefore least stable energy, corresponding to the anti-bonding level). In passing, we can just note that EA - EL = 2~. On inserting EA into eqns (IV) and (V), we obtain (cj + C 2)~ = 0, that is CI = - C2 = CA . In the same manner, on placing EL into eqns (IV) and (V), we discern (- CI + C2) ~ = 0, that is to say CI = C2 = CL . The wavefunction of the Hi system will allow us two solutions which can be written in the following way: \ilL = cd
\ilL =

.J2 (
\IIA =

.J2 (
I

-
378

Optoelectronics of molecules and polymers

As shown in Figure 6, a schematisation of these wavefunctions is quite possible. It is of interest to see that for the bonding solution, the electrons displays a very high probability of being between the two nuclei so that the strong electrostatic attraction between the electron and nuclei (I) and (2) bonds the entire system. However, we can see in the anti-bonding system that the electron displays a high probability of being on the outside of one of the other nuclei so that the strong electrostatic repulsion between nuclei (1) and (2) is destabilising. Comm ent

With S = <
=1= 0, a similar, longer calculation results in: III _ L -

III _ A -

EL=---

EA=---

<X + ~

I+S

<X -~

l-S

b The molecule H2

In H2 there are two electrons which we can assume are placed at the same potential as the electron in the previous exampleof Hi ' as to a first approximation we can neglect the interaction potential between the two electrons. Each of the two electrons has availableto it two orbitals (IlIL and III A)' The system with the lowest energy would be with both electrons each in a bonding orbital. In this 'l'L. :

•

..

Wavefunctions

/~

I . '" ~ ~ • • ~..

CP.I ,

I"

•••••'

6 ' I '

• _,2

_

/

_.'

2

Probability of presence

....-

---

......- - -

, \,

,

1 "

•

\~

} 'A

__z

~ ,.. ,-- -
.

.

Bonding and anti-bonding orbitals (equi-energy curves)

Figure 6. Wavefunctions and orbitals in Hi: left and right, respectively, bonding and antibonding orbitals.

A-I Atomic and molecular orbitals 379

<;"

EA

<;'.

~@ ~

~~ i----:'--«'----=-..l~ ~d" Independent hJdrOJ:en atoms denoted I and 2

Degenerate enel'J:Y level for as) . lem of 2 independent 01001

;::

Bonding and anti - Prohabllity of presence bonding energy densities for bonding I odenslty surfaces for: (a) antilevels as aloms I and unti-bondlng states and 2 come close bonding orbital; and (b) bonding orbital

Figure 7. Energy and electronic states in H2. state, with the electrons localised around the position between the two nuclei, the system energy would be minimised as the electrons would benefit from an attracting potential (negative potential energy) due to the overlapping of Coulombic potentials developed by the positively charged nuclei . This charge localisation (and the maximum electronic density between two nuclei) is the very basis of the concept of covalent bonding. When looking at the overall system, the combination of two hydrogen atoms resembles the melting of one into the other with the final result of an increased local probable electron density in between the two nuclei . The bonding state is again realised using a linear additive combination of the individual orbita ls of each atom , as shown schematically in Figure 7. The wavefunction can be written, generally speak ing for both bonding and anti-bonding states , as \II = c\ (jJ1 + C2(jJ2 and result s from the linear combination of atomic orbitals (LCAO) .

Comment

-

While the function

\IlL

is symmetric, the function \II A is antisymmetric.

on the EL level, the spins are anti-parallel so as to distinguish the two electrons (undistinguished from the point of view of the orbitals as \IlL remain s unchanged if they are permuted) so S = 0 (singlet states); on the EA level, the spins can be parallel as they are distinguished at the level of the orbital (\II A changes sign if the two electrons are permuted) so S = 0 or I (singlet or triplet states).

decreasing

potential Is

Is

energy

Figure 8. Is (J- and (J* -orbitals.

380

Optoelectronics of molecules and polymers

A

D

Figure 9. 2p (J- and (J* -orbitals.

3 a- and n-bonds a a-bonds Sigma (a) bonds are those which have as axis of symmetry the straight line in between the two covalently bonded atoms. For example, from a pair of s-orbitals we can derive the (o-orbitals shown in Figure 8. If the axis Ox is that on which the two atoms (A and B) are placed, the 2p orbitals will have the combination shown in Figure 9. With 'lJL = ~('lJA2Px + 'lJB2px) and 'lJA = ~('lJA2Px - 'lJB2px) '

A

B ()

AD

~

b

~

o

1t·2p.

1t 2P.

A-I Atomic and molecular orbitals

381

b n-bonds n bond s are molecular orbitals which have a plan of symmetry which passes through the straight line Ox joining the atoms and is perpendicular to the direction Oy for n2pz-orbitals or the 'direction Oz for n2py-orbital s, as shown in Figure 10.

II The covalent bond and its hybridisation 1 Hybridisation of atomic orbitals

a What is hybridisation? Imagine a carbon atom with the atomic configuration IsZ2s z2 pz. While it should behave like a divalent element, as in COz, the same configuration does not explain how CH4 comes to exist in its tetrahedral sp3 state , how CHz = CHz exists in its triangular hybrid sp3 state nor how CzHz presents a diagonal sp! hybridisation. These structures can be interpreted on realising that when two carbon atoms are 'united', the valence states of each carbon atom (2s and 2p) are in effect presented as one s and three p orbitals. This first excited configuration (l sZ2 s 12 p3) is favoured if the energy levels of its constituent orbital s are lowered on its formation, and this is what happen s when they are 'mixed', or rather, hybridi sed. The general form of molecular orbital s derived from the process of hybridisation can be written using quite self-evident notations (here carbon atoms A and B are identified by I and 2, respecti vely): I\II} =AII2s}\ + AzI2px}' +A312p y}( +~ 12pz }' + B 112s}z + Bzl2px}z + B31 2p y}z + B412pz}z = I\III } + l\IIz}·

b Different degrees of hybridisation There are indeed different levels of hybridi sation-different levels of coupling of each of the various and involved orbitals. Diagonal spl hybridisation Px orbitals gives:

This sort of hybridisation, involving for example s- and

I\II} = AI12s}, + Azl2px}1 + BII2s)z + Bzl2px}z = l\IIl } + \\IIz }.

in which I\II d and I\liz} corre spond each to 2 possible orbital s to give a total of 2 bonding cj -orbital s and 2 anti-bonding -orbitals, So I\III} can thus correspond to l\IIll} = aIl2s}I ± azI2px}' or to l\II la} = a; 12s}, ± a;12px}\ and in both cases the + and - give the signs of each of the pairs of hybrid orbital s. The resulting n-orbitals are obtained with respect to both Oy and Oz, as shown in Figure II.

a:

382

Optoelectronic s of molecule s and polymers

7t"2p, 2 anti-bonding n· ·orbitals and 2 bondi ng n-orbi tals

7t 2p, 7t"2p, 7t 2p, Figure 11. rr" and rt-orbitals.

Triangular sp2 hybridisation The system couples only the states 12s > , 12px} and 12py} while leaving the state 12pz} outside of this linear coupling. The coupling can thus be written:

+ A212pxh + A3/2p y} I + Btl 2sh + B212pxh + B312p yh

\II = A112s}I

= I\III}

+ 1\II2}

The three hybrid orbitals have their axes in the same plane and are generally denoted 2spi , 2sp~ and 2sp~ . In a plane perpendicular to these is the fourth orbital, 2pz, which gives rise to the rr-orbit al, For the molecule C2f4 below, the orbit als are schematised in Figure 12.

H ---..C ==C/H

H---

~

The angle s between the three hybrid a-orbitals, which originate from the same carbon atom, must be 120 o . A calculation performed in the same manner as that shown in the following Section 2 (for the sp3 hybridi sation), allows a determinations of the three

Figure 12. Orbitals in the molecule C2H4.

A-I Atomic and molecular orbitals

383

hybrid orbitals with respect to the sp2 hybridisation :

Tetragonal hybridisation This state draws on all states without exclusion through a hybridisation which corresponds to the structure of diamond and is further detailed below.

Sp3

2 sp3 Hybridisation a Rotational symmetry of the orbitals The normalised wavefunctions for the s- and p-states (with r ~ rf) of the valence electrons in carbon (n = 2) result from initial calculations (see Section 1-1): Is) = R n,o(r)80,o¢o = h(rrJ = S IPz) =
~2'1; IIR(rr) cos e = g(r) cos e

= zf(r) = Z =
(n = 2, 1 = 0, m = 0)

(n = 2, 1= 1, m = 0)

~2'1; IIR(rr) sin ecos
= g(r)sinElcos

(n = 2 , I = I , ill = I)

~2'1; IIR (rr) sin osin
= gtrj sin Osin c = yf(r) = Y =
(n = 2, 1 = I , m = -I)

(the functions fer) and g(r) are related simply by g(r) = rf'(rj) . The four proper functions, S, X, Y and Z constitute the orthonormalised basis for a space in four dimensions. On considering the carbon hybrid state sp3 in which none of the four orbitals (X, Y, Z and S) playa specific role, the proper functions of this new state can be developed using the preceding basis, although we now have four more functions to deal with given the number of dimensions in the system; lll i =CltS+alX+blY+CIZ

j

1lI2 = Cl2S + a2X + b2Y + C2Z 1lI3 = Cl3S + a3X + bj Y + C3Z 1lI4 = Cl4S + U4X + baY + C4Z

384

Optoelectronics of molecules and polymers z

x

Figure 13. Representation of IlJj .

IWI} , IW2}, IW3} and IW4} are the hybrid orbitals which correspond to the excitation 1s22s22p2 --+ 1s22t4. This is only possible if the reached energetic states are equal to the gain in energy required by the atomic excitation. Here m is the unit vector going in the direction and the point M has the spherical co-ordinates rr, eand <poIn Cartesian co-ordinates ex, ey and e z are the unit vectors for ----+

.

the axes Ox, Oy, Oz, and the respective components of OM are x = rf sin e. cos
e

Given the form of X, Y and Z, detailed at the beginning of this Section, we can write : x = g (r) . ~x . ~ Y = g(r) . ey . m Z = g(r) . z .

[

e m

We can now write (with i = 1,2,3,4): Wi = ajS + g(r) (aje x + bjey + cje z) . m. On introducing the vector tj = aiex + bjey + Cjez , which thus has components ai, b., and Cj in the system based on the axes Ox, Oy, Oz. Each function Wj can now be written as: Wi = cqS + g(r)tj . m. With Iml = I, Itil = Jar

+ br + cr and on setting co =

(tj, m), we have

We can state that Wj = ajS + ~j , in which the orbital ~j thus introduced exhibits the form ~j = a.X + b.Y + CjZ = g(r)tj . m. Once ~i is normalised co-linearly to ~i, and by consequence verifies: ( ~i1 ~ i) = 1 and ~j = A.j~j.

A-I Atomic and molecular orbitals 385 ~i

thus takes on the form

~i

-, -, ---+ ---+-, = g(r)ti . m = g(r) cosu], m) = 00 . m, in which OD = g(r)t j •

= (t~ , m), now ~i = OD cos 00, the position of points M' are such that ---+ ~j = OM' is a sphere with diameter 1001 = g(r) and is directed following the unit vector ti . Figure 14 shows, in the plane of the paper, a representation close to that which allowed us to determine the pz orbitals. Similarly, the M points are placed so that OM = ~i = Ai~I giving rise to a sphere of radius Ai 00. As in Figure 13, we finally reach the hybrid orbital Wi through the addition of the S orbital (which has a spherical symmetry around 0, as shown in the Figure) with the orbital ~i = Ai~i, and is also equivalent to a rotation about t~ or about ti(= Ait~). Using

00

-,

b The angle between the direction of equivalent hybrid orbitals It is possible to determine the expression for the angle between the directions for two orbitals of the ~j (or ~i) type in rotation (so that the WI and W2 orbitals are equivalent). z

o '*_-'---_~L.-.y

x

.........

Figure 14. Representation of
386 Optoelectronics of molecules andpolymers If we take the indice of the two orbitals under consideration using eqns (I) and (2), we can write, for example, IIII

= (XIS +;1 = (XIS +

W2

AI~I

= (XI(S +

J.LI~I)with

AI

= (XtJ.Lt;

= (X2(S +

J.L2~2)with

A2

= (X2J.L2.

= (X2 S +;2 = (X2S +

A2~2

The orthogonal addition of IlII and 1lI2 gives: (1lI111lI2)

= (XI(X2(S + J.LI~I IS + J.L2~2) = (XI(X2[l + J.LIJ.L2(~11~2)] =

°

(with the functions S and ~i orthonormal with respect to one another). With (XI and (X2 unequal to zero, we can say that we should have 1 + J.LI J.L2 cos (X12 = 0, in which (Xl2 = ct~, t~) represents the angle between the two directions given by the rotational axes ofthe two normalised wavefunctions ~l and ~2. The interchangeability of orbitals applied to the orbitals IIII and 1lI2 in effect impose that J.L I = J.L2 (Ill I and 1lI2 orbitals, as for S orbitals, are equivalent as the parenthesis has the same coefficient of I, and so we should have J.L t = J.L2 = J.L so that III t == 1lI2). Finally, we can state the orbitals III t and 1lI2 are equivalent if cos (X l2 = - ~. fL

c Wavefunctions in the sp3 hybridisation

Here we now have a carbon atom at the centre of a tetrahedral with orbitals about axes that join the centre to the extremities of the tetrahedron, as shown in Figure 15. The lines of the axes have angles between them which can be calculated, for example, using the t~ and t~ components. As It~ I = 1 and the carbon is at the centre of a tetrahedron, we should be able . that a'j 2 + b'12 + c,j 2 = 1 and a' i other terms that a' = to wnte a l = b'I = c,I' and ill at a, b~ = c~ = ~. Similarly, for t~, and in taking into account its geometrical position, , , I db' I a2 = c2 = J3 an 2 = - J3' From this we now have cos (X l2 = t~ . t~ = that is (X l2 = 109°28' and - ~ =

-1,

-1, which is the same as writing J.L = ../3.

Figure 15. sp3 Hybridisation.

A-I Atomic and molecular orbitals

387

Using \lJ] = aleS + Itl~]), we can say that \lJ, = al(S + J3~I) , and the normalisation condition for \lJ I brings us to a value for a I:

Finally,

in which (ai, b'l' ci) are the director cosinus : ai = bi = c', = ~ , to yield :

We obtain :

\lJj = 1(S +
1

\lJ4 = 1(S -
+Y -

Z)

A-2

Representation of states in a chain of atoms

I A chain of atoms exhibiting a-orbital overlapping 1 o-orbitals and a compliment to the exampleof 8 atoms in a chain A chain of 8 atoms (N = 8) was detai led in practical terms in Chapter I, Section VII. Each atoms presented was considered as having the s state . The energy levels are shown in Figure 1-24 and the repre sentative functions for the lowest lying band bonding states (of the function 11114 ) and the highest band antibonding states (of the function 1111<0) are shown explicitly in Figures 1-22 and 1-23, respectively. Figures I-a and l -b describe the forms of these func tions as an aide -memoire. In addition , between k = leo and k = k.t there are the intermediate states, which are represented the real parts of the wavefunctions III~ I and 1II~2 for k = kl in Figure 2 and for k = k2 in Figure 3, with 8

L cos(kpta)wo(x -

lII~p (x) = R (lIIkp(x) = Co

ta).

1=0

Here, I

Co = -

(2:

P

VN' kp =

2n: Na

and

1II0(x - ta) = Ce- crx •

For p = I, we have: n:

k, = -

4a

and

2n: Al = - = 8a. kl

The successive values for t, k , ta = ~ t, and cos ~ t are listed in the table below. 2

3

4

n

rt

4

2

3n: 4

rt

0=8 k jta

= n

cos - t 4

n: 4"t

0

.,fi

-

2

= 0.707

5 5n: 4

6

7

3n: 7n: 2

4

0 -0.707 - I -0.707 0 0.707

390

Optoelectronics of molecules and polymers

........ljlss"i.(jlso

(a)

.----+-----.----+-----.---...... --____4~-- .......--____4I~___+x I L=Nd

1...

..;(j

::iji;.j

~·i···· ·· · · ·:·ip;j· · · · · · ··· · · · · ·iP.

::..ip;;

iji;.··········:··ip;;·············ip;i ;;iji;o····· (b)

Figure 1. (a) Representation of IJlko =
\IJ~I = \PsO",8 + 0,707 . \PsI - \Ps4 -

+ 0 . \Ps2 - 0,707 . \Ps3 0,707 . \Ps5 + 0 . \Ps6 + 0,707 . \Ps7

We can thu s see that when k = k j, the nodal points start to appear in the wavefunct ion as shown in Figure 2.

.----------,::>"""''''"---..., "

,,

,,

I

,,

I

I I

,

,/

'" ........... ....

1.., = 8a

_--

__ I

I

..

;'

"

Figure 2. Representation of IJIkl with IJIkl = Re(lJIkl) =
+ 0 .
A-2 Representation of states in a chain of atoms

391

- For p = 2, we have

Jt k2 = - , 2a

and

2Jt ).2 = TI = 4d 2a"

The successive values of t, k Ita = I t, and cos I t are shown in the table below:

0=8 kjta =

I 2

3

4

5

6

7

Jt Jt 3Jt 5Jt 7Jt 0 = 4Jt - Jt - 2Jt - 3Jt 2 2 2 2 2

-t

o -1

0

o

Here we can state that at the centre of the band, for k boding nor antibonding, as shown in Figure 3.

- I 0

= k2, the states are neither

2 General representation of states in a chain of overlapping (J s-orbitals When we have positive s-orbitals (a lone, single atom with an s state-see Appendix 1-4, Figure 4) and a chain as in Chapter I, Section VII, we have from Chapter I, eqn (20) -

~ s = N/o(x - ta j] W(x - sa) l\{Io(x - sa)} = (\{It I W I\{Is) < 0, as \{It and \{Is have the same positive sign (from the orbital of the lone nOt or nOs atoms) while W < 0.

From an energetic point of view, the result with E given by eqn (22) of Chapter I that E, = Eos - a s - 2~ s cos ka, and the energy is, as in the lower half of Figure 4 for s-orbitals:

k

A=4a

<:..fi"~5·-·· -·---·--~··]'"'--·--····· -1s".::.::.::.

'\1 I V ...............

Figure 3. Representation of 1lJ~2 with 1lJ~2 4ls4 + o· 4ls5 -- 4ls6 + 0 . 4ls7 ·

....i.4-Re(expiikjta)

...........

= Re(llJk2) = !PsO~8 + O· !Psi -

4ls2 + o· 4ls3

+

392

Optoelectronics of molecules and polymers E

.--.-........' - 2~""

E~ ~O'*-Pb~d

- p band E~--f"2jtaP--j

,

-Ebs--t-

i

Os

- 1tia

;

!

;

0' -

Os -

t

Msban1 t s- - -

* - s barid

,

I

2~s

i

rita

Figure 4. E = f(k) curves for the a overlapping of sand p orbitals.

•

•

lowered at the lowest part of the band as -~s is always negative and cos ka > O(-rr./2 ::::: ka ::::: rr./2 at the bottom of the band) and we have a bonding state (c-s band); increased at the highest part of the band as -~s is always negative and cos ka < O(n 12 : : : ka ::::: rr. at the top of the band) and we have an antibonding state (o'*-s band).

This result can again be found by taking into account the phase term associated with each wavefunction for each atom in the chain : •

•

two adjacent and bonding s-orbitals are in phase (k = 0 as in Figure l-a) and have the same sign . The geometrical form and the orbital and its sign can also be shown graphically, as in Figure 5-a. The interaction (coupling) of the two states gives rise to a decrease in the energy (the o-s band); however, the antibonding states are associated with adjacent orbitals which have opposite phases (when k = ±rr./a) and also therefore opposing signs, as in Figure l-b. This situation is represented in Figure 5-b in which the geometric form of the orbitals along with their signs are shown . The energy resulting from their interaction is increased (in an unfavourable state) and the result is the (0'* -s band shown in Figure 4.

(aJ~
k =0

(bJ~------+e------~@------ie--------t@

k = rda

Figure 5. Distribution of: (a) a-s bonding orbitals; and (b) o"-s antibonding orbitals (arrows indicate phase terms).

A-2 Representation of states in a chain of atoms (a)

00

393

-~------ -~-------~-~ x -~----- --~----- - -~-~

Figure 6. Distribution of: (a) antibonding 0 * -p orbitals; and (b) bonding c- p orbitals (arrows indicate phase terms).

3 General representation of states in a chain of overlapping cr p-orbitals For the p-orbitals, once again we can use -~ap = (\Ilo(x - ta) IW(x - saj ] \I1o(x - sa) ) = (\IltlW I\Il s ) where W < O. Now the functions \Ill and \Ils represent p-orbitals, which have the form shown in Figure 6. Noticeably each has adjacent positive and negative lobe s so that -~ap is positive (see the upper half of Figure 4 which details a-p orbitals). We can note that: •

when -n /2

:s ka :s n /2 , cos ka >

Ctp ;

•

when n/2 :s ka :s nand -n :s ka decreases by E~p = Eop - Ctp.

0 and the energy is increased by E~p = Eop -

:s -n /2, we have co s ka <

0 and the energy

The same result can be found : •

•

when k = 0, the wavefunctions are distributed in phase (following Floquet' s calculations, the term cos kpsa is equal to I ) and they can be geometrically represented as shown in Figure 6-a (using as example the px-orbital s). A positive lobe interacts with a negati ve lobe to produce an antibonding state with the appropriate increase in energy (o"-p band shown in Figure 4). Thi s beha viour is in opposition to that of the s band in the same k region; or when k = ±n/a, the phase factors altern ate (cos kpsa equ als, alternatively, + I and - I) so that the geometric repre sent ation is that shown in Figure 6-b and corresponds to the interaction s between lobes of the same sign. The result is a decrease in the energy for what is a o-p bonding state as detailed in Figure 4.

II n Type overlapping of p-orbitals in a chain of atoms: n-p- and n*-p-orbitals The pz-orbitals bonded to each other by a n -bond give rise to a n - p type orbital, as shown in Figure 7. The resonance integral , -~ llP ' involves both the overlapping of the two positive and the two negati ve lobe s. With W < 0, -~ Jtp is now negati ve (as for - ~s) giving a behaviour similar to that of (a -s-bonds . Figure 8-a and -b show, respecti vely, the geometrical representation s of the lowest energy bonding and highe st energy anti-bonding orbitals.

394

Optoelectronics of molecules and polymers

z

Figure 7. The rr-p-orbital,

(a)

(b)

Figure 8. Distribution of (a) n- p-orbitals and (b) rr"-p-antibonding orbitals.

III

(J-S-

and o-p-bonds in chains of atoms

In general terms, and as we have seen in Chapter I, with k = n j a a stationary wave system can easily arise. The wavefunctions and the presence probabilities take up solutions of the type (Figure 1-5) in which: • •

the electron charges are esse ntially spread in the neighbourhood around the lattice nodes, so that \11+ ex: cos( n xja) and p+ = cos 2( nxja); and the charges are essentially about the mid-point between lattice nodes, so that \11- ex: sin( nxja) and P- = sin2( n xja) .

Given the geometri cal shape of the atomic s- and p-orbitals and their corresponding electron distribution s: • •

the electron charge s associated with s-orbitals are concentrated about the atomic nodes with probability p+ ; and the concentration of electrons associated with px-orbital s, mid-way between atomic nodes, can be expressed as the probability P- .

In terms of energy the two orbitals and their associated electron concentrations are separated by a gap of EG which is precisely defined by k = n j a. In addition, as the waves (\11+) associated with the s-states tend to concentrate electrons in the neighbourh ood of the centre of the atoms, the electrostatic interaction energy effectively lowers the energy of the electrons, which are more stable than

A-2 Representation of states in a chain of atoms

395

those associated with the progressive wave (in exp(ikx)) which delocalises any sort of electron. Conversely, the \11- wave associated with the p-states concentrates electrons at the mid-point between atoms (that is the further point from the centre of the atoms involved) and minimises as much as possible the electrostatic interaction energy. The result is that the electron charges are less stable than those associated with a progressive wave (in exp(ikx) delocalising all electrons) of the s-states \11+ wave. Thus, in this particular example, an energy gap can appear in a I-dimensional system with the simultaneous existence of at least two orbitals (here sand p). Figure 4 can be compared with Figure 1-12.

IV Comments 1 The Bloch function In Chapter I it was shown how with a zero order approximation (an electron placed in a flat-bottomed well) an electronic wavefunction can be written: \11k (x) = \110 = e ikx

(to the order of a normalisation constant).

However, for a semi-free electron (placed within a potential well with a bottom perturbed by electrostatic interactions with atomic nuclei situated at lattice nodes and x) we need to find a wavefuncundergoing a periodic perturbation w(x) = wo cos tion for the perturbed state of the type \11k (x) = \I1ou(x) = eikxu(x), in which u(x) is the unknown function . With a potential perturbation of w(x) , the Hamiltonian can be written as 2 d H= dx 2 + w(x) ,

2;

-n.

and is conserved on undergoing the translational operator Ta, which transforms x into x + a. This can be expressed in TaH =

_n.2 d(x +d a) 2 + w(x + a) = -n2 -dxd 2 + w(x) =

H.

H remains invariant with respect to the effect ofTa, and commutes with Ta: [H, Tal = O. We can thus find the same proper system of functions for T, and for H, in a process which is followed up in most standard text books on quantum mechanics. In order to find the form of the proper functions of Ta which can also be taken as the proper function of H, we can use the following equation which they verify :

The product from the application of these functions is T a 0 T - a, and involves successive translations with modules -a and then a so that the operator T, 0 T - a changes

396

Optoelectronics of molecules and polymers

as though it has the identity I. Thus:

From this we can deduce that Ck,aCk, -a the proper values becomes:

= I, that is Ck,a = eika, and the equation for (1)

From outside we have the definition of the operator Ta which transforms x to x + a, and allows the expression:

(2) On comparing, eqns (1) and (2) we can extract u(x) = u(x + a). So, the proper function Wk (x) = eikxu(x), which describes the state of a semi -free electron-which has the form under analysis-is such that u(x) = u(x + a). This proper function, Wk(X) = eikxu(x) with u(x) = u(x + a) is called the Bloch function.

2 Expression for the effective mass (m") By definition, m * = Fex! , in which y = ~ and vg, in the dualistic theory, is given by the velocity of the package of the wave group as dw

vg

= dk '

so with E = lu»,

IdE

vg - lidk'

and

Yg

=

ld (dE) lidt dk

2E

=

Id dk Iidk 2 dt .

In addition the exterior work force (Fext = qE ext for an external field Eext applied on a charge q) is such that dE = Fextdr = Fextvgdt, so F

ext

=

IdE vgdt

=

IdE dk vgdk dt

=

d~ dE dk dk dk dt

= Ii dk . dt

Calculating the ratio Fext

Y

2

. * 1i = m* grves m = d2E' (jj(2

For a free electron, E

= n;~2 , and we find again m = m* as Fint = 0, so that Fext

= FTotal = my = m*y .

A-3

Electronic and optical properties of fullerene-C60 in the solid (film) state

I Electronic properties of fullerene-C60: the band scheme as indicated by the Local Density Approximation Method [Tro 92 and 96] Figure I schematises the first Brillouin zone of the cubic face centred (cfc) system of fullerene-60 (C60) . Table I goes on to detail the energies of states at the centre (I') of the Brillouin zone and have been calculated using the density function in the Local Density Approximation (LDA) [Tro 92]. In Figures 2-a and -b the theoretical state densities are compared against empirically obtained photoemission or inverse photoemission spectra, respectively. In Figure 2-a, the theoretically obtained spectra of the density of full states, compared against photoemission spectra, and the highest energy peak arising from a valence band results from the TIs-states. These states are derived from the HOMO h u k,

x

ere

Figure 1. Brillouin zone for a efe structure.

398

Optoelectronics of molec ules and polymers

Table 1. Electronic state energies for C60 at the Brillouin zone point

r.

Energy (eV)

Labe l

Energy (eV)

Labe l

Energy (eV)

Labe l

-18.766 - 18.146 - 17.262 - 17.22 1 - 16.098 -15.381 -15.326 -14.137 -13.983 -13.585 -13.542 -12.021 -11.993 - 11.873 -10.91 7 - 9.830 -9.752 -9.564 - 9.423 - 8.252 - 8.128 - 7.368 -7.271 -7.114

aO-ag a , -tu a2- tg

-7.109 - 6.533 - 5.829 - 5.802 -5.563 -5.389 -5.371 -5.257 -5.004 -4.946 - 4.929 -4.587 - 4.109 -4.089 - 3.597 - 3.452 - 3.189 -3.120 -2.766 -2.755 -2.628 -1 .850 - 1.615 -1.084

a7- tu a7- tu a7- au a7- tu IT,-tu

-0.965 -0.225 0.000 1.710 2.372 3.469 4.052 4.103 4.217 4.971 5.149 5.515 5.878 6.240 6.423 6.495 6.974 7.212 7.430 7.454 7.815 8.298 8.318 8.754

1q -tg

a2-eg a3- tu a3- au a3- tu a4-eg a4-tg a4- ag a4-tg as-eu as-tu as- tu as-tu a6-tg a6-eg a6- tg a6-ag a6- ag a6-tg a7-eu a7-tu ITO-ag

IT2-tg as-eg as-tg IT2-eg as-ag as-tg as-tg ag-tu ag-au IT3-au IT3-tu as-eg as-t g ag-eu ag-tu IT3-tu IT4-eg IT4-ag IT4-tg

ITS-eu ITs- tu ITs-tu IT6-tg ITs-tu IT6-eg IT6-tg Co-ag IT7-eu IT6-ag IT7-tu ag-au IT6-tg ag-tu OO-ag TOA-au TOB -ag C, -tu alO-tg ag-tu C2- tg alO-eg O,-tu

level of the molecule, which is however broken down on the interior of the tetrahedral crystal to give 5 bands with symmetries tu and e u. Yet this dispersion of bands does not permit their resolution and the breakdown can only be discerned at the point r (see Table I) . In the same Figure 2-a, the band of TI4 -states derived from the molecular states gg and hg. On increasing the energy of the bond , there is the first appearance of (J and TI charac ter mixtures. In Figure 2-b, the theore tical spectral state dens ities calculated for empty states are compared against inverse photoemission spectra. The conduction band, here called 'A', is formed from TIS states, which are derived from LUMO states exhibiting T,u symme try. There follows this the band ' 8 ' formed from TI6-states and appears at an energy lower at the point r than in the state densi ty spectrum. On dissecting the various contributions which go up to make the 'C' band, it is possible to see that the first non-mo lecular state is at 4.217 eV. This state and its wavefunction are localised about the centre of the molecules and, characterised by L = 0, is denoted Co.

A-3 Electronic and optical propertiesof fullerene-C60 in the solid (film) state

E,=3225 eV

....'"

'" .:!:!

'tl

=

-e

....t'l .... rJ:J

.... .... rJ:J

,~

399

b

.<;:

';;

'" =

~

~

~

~

~

-20

-10

Energy (eV)

o

o

10

Energy (eV)

Figure 2. (a) Calculatedstate density functionsfor the valenceband and empirical photoemission spectra. The lower 2 plots show density distributions for (J and IT at I'd of the Brillouin zone (bands numbered in terms of L); and (b) the lower conduction band function densities compared against two experimentally obtained inversephotoemission spectra.

Using the self-consistent field theory it was shown that there are relatively flat regions at the centre of the C60 sphere and in the tetrahedral and octahedral interstitial sites of a cfc lattice. The wavefunctions localised around these sites appear with energies between 4.8 and 5.3 eV above the conduction band and are called 00, TOA and TOB, with the latter two names being, respectively, for bonds and antibonds around a tetrahedral site. The D and E peaks are due to simultaneous contributions from both molecular and non-molecular states; in effect they have reached the stage of delocalised states . We can see that with these results the derived peaks are distinct, considerably more so than those observed for graphite and diamond and would tend to indicate the molecular nature of solid C60. As already discussed in Chapter Ill, we can safely say that Hiickel's method results in a good approximation. To end this Section, we have shown the structure of the C60 crystal bands along the ~ senses in Figure 3. They are defined by the points r and X from the Brillouin zone and Z defined by X and W, as shown in Figure I. Figure 3 shows energies close to the fundamental gap and shows the highest group of valence bands and the two lowest groups of conduction bands . The direct gap is here determined as 1.2 eV at the point X, although this method tends to underestimate the actual value .

400

Optoelectronics of molecules and polymers

---=::

3.0

-.

1T6~ 2.0

:>

1T,

Col '-'

~

"-

LO

Col

C

W 0.0

:>:<

1T,

~ -LO

r

x

z w

Fig ure 3. Band structure of C60 in a cfc structure along the 6. and Z directions .

t 1u

'i ·· ·:,,v

LUMO

·, ,

HOMO

(b)

Figure 4. (a) The relationship between the experimentalI y derived state densities in the solid and the molecular orbitals levels along with permitted transitions [Law 92]; and (b) permitted transition s derived from the energy levels of molecular orbitals. The v band is forbidden for reason of symmetry, however, it is not excluded from electron-phonon coupling effects and appears weakly in the spectrum [KeI92].

A-3 Electronicand optical properties of fullerene-C60 in the solid (film)state

401

II Optical properties and observed transitions It has been observed that the absorption spectra derived from solid C60 were very close to those of C60 in the molecular state , a result which indicated that C60 displays only very few interactions between molecules in the solid state [KeI92] . The eventual similarity of spectra derived from solid and solution state C60 allowed an attribution of the solid phase bands, along with help from known optical transitions associated with the energy diagram of molecular orbitals as shown in Figure 4-a. Even though the h u ~ tl u transition is forbidden, (as it accords t.L = 0 and not t.L = ±I-see Figure III-I 3 where the h u and tl u levels both correspond to L = 5), it is nevertheless just observable (Figure 4-b) . This is due to vibronic coupling which modifies the selection rule (see Chapter VII on optical properties for further details). And accordingly a rather weak band at 550 nm (2.50 eV) was attributed to this transition [Law 92] with an energy rather elevated when compared to alternative estimations (from 1.5 to 1.9 eV). Figure 4-a permits the deduction of the energy of additional transitions which are h u ~ tl g, hg ~ tl u, h u ~ hg, gg ~ t2u , and h g ~ t2u,each of which has, respectively, an energy of 3.0, 3.4, 4.3, 5.0 and 5.4 eY. The se values coincide quite closely to those experimentally derived [Kel 92].

A-4

General theory of conductivity for a regular lattice: the Kubo - Greenwood equation for conductivity in delocalised states and expressions for mobility

I Electron transport effected by an external force and its study 1 Effect of force on electron movement and reasoning within reciprocal space Here we take as example a square plane crystal with sides (L) defined by L = na. The Brillouin first zone and the dispersion curve (E(k)) exhibit the forms represented in Figure I . The first band is assumed to be partially occupied by electrons in a symmetrical manner so that E(k) = E( -k). At the point of thermal equilibrium, the velocity of an electron can be given by :

_ l~ v = -gradkE. Ii

(I)

As all the carrier velocities cancel each other out (orthogonal in the circle E = constant as in Figure l-a) the total result is zero and there is no charge tran sport.

E

- rc/a;;-l--I-'--t--+-+-~·kx

- rc/a - rc/a

rc/a

Figure1. Representation of a reciprocallattice with: (a) the Brillouin zone and an equi-energy surface; and (b) the dispersioncurve E = f(kx ).

404

Optoelectronics of molecules and polymers

Comment Eqn (1) can be justified by the following: the velocity of an electron can be repre sented by the velocity of the group associated with the wave packet, which describes the wave-like character of an electron: I aE

aw

with E

= lu» ==:} vg = h ak '

I ----+ _ vg = hgradkE = v

(simplified notation ).

vg -' - ak' to give in 3-D:

If an external force (F) is applied for a certain time (dt) on an electron which has a velocity then the resulting change in energy is expressed by dE = F. dt.

v,

v.

Following eqn (1) we have: dE = Additionally, in 3-dimensions

F(kgradkE) dt.

By identifying between the two last equations,

-

dk

F=n-. dt

-

(2)

-

Eqn (2) shows how F and dk are colinear and result s from the e!fect of a force displacing an electronic population in the Brillouin zone parallel to F (dashed circle in Figure I) , and the distribution symmetry is perturbed (as E(-k) =1= E(k)) . The resulting speed of the electrons is non-zero and charge transport is initiated .

2 Boltzmann's transport equation At the thermodynamic equilibrium, the probability that an electron with a wave vector k will be in the reciprocal space is given by the Fermi-Dirac function:

fo(k) =

I ~ 1 + e(E(k)-EF) /kT

(3)

This function changes in the presence of an external force: f (k, r, t)

= fo(k) + fl (k , r, t)

(4)

Most generally, this function depend s not only on k but also on the geometrical position (r) of an electron, when the force induce s a local inhomogeneity in temperature or concentration, and also on the time (t), if the permanent regime is not reached. During an interval of time dt, the variation in f is equal to the sum of variation s caused by the applied force (interactions with the exterior) along with a contribution

A-4 General theoryof conductivity for a regularlattice

405

due to collisions (internal interactions). This gives:

M(M) = dt

dt

applied force

+

(M) dt

collisions

And with (df)rorces

= (Oinstantt-dt -

(Oinstantt

f(k~ ~r) - f ,

(~dk k - -dt ~r -

I

=-

= -[(Oinstantt -

dt

dr ) -dt dt

'

I

(Oinstantt-dd

of = - -dk ok

of -dr

or'

during the interval [t - dt, t] we have: df ) of dk ( dt forces = - ok dt

-

of dr or dt '

and in 3-dimensions: -

dt

dk

-+ (fr - grad.f . -

dt

dt

-+ = -gradkf . -

df )

(

forces

On introducing the last expression into eqn (2) and given that df ) (dt

(r, k)

-+ = -n- l~F · gradj f -

v=

-+

~~, we find:

~-+

v . grad.f

Applied force

With respect to the collisions, the contribution is in the following form (system with a relaxation time): (

df ) = _ f - fo, dt collisions t

*

t

being the relaxation time .

Under a permanent regime, is zero and we find Boltzmann's equation (in which we have replaced f by fo for all gradj f and grad, f , in a legitimate act as fl « fo and the perturbation is weak and therefore negligible with respect to the value at the equilibrium state). I~

-+

~

-+

Ie F · gradj fo + v . grad, fo +

°

.

-+ homogeneous, then grad.fi, = 0, and

r

If we assume that the crystal we have eqn (5) becomes -+ n- I~F · gradkfo + -fl t

=

f

-l

IS

(Boltzmann 's equation).

I~ -+ = 0, or even f, = -tli- F· grad; fo

(5)

(5')

On taking the latest expression and inserting it into eqn (4) f = fo + f], and on using eqn (I) for velocity to finally obtain

v = *grndkE, we can write that grndk fo = ~gradkE = n~~ v, f = fo -

t .

~

~

ofo

F.v. ClE

(6)

406

Optoelectronics of molecules and polymers

II State density function, carrier flux and current density in the reciprocal space 1 General expressions for fluxes of particles The flux of charge carriers generated by an applied force (F) is consi stent with respect to the product of the concentration (c) of charge carriers of a certain charge (q) and their displacement velocity (Vct). This is because the flux repre sents the quantity of charge which traverses a unit surface in a unit time. So we can write for the average over all carriers that (<1» = (cqVct ). In reciprocal space , the carrier concentration is expressed by c = Jk N(k)f(k)d 3k, in which N(k) is the state density function and repre sents the number of possible states for carriers in a unit volume in reciprocal space when they are in a unit volume in direct space . We can thus write that <1>

=

1

N(k)f(k)q(k)v(k)d 3k .

(7)

2 Expressions for the state density function a Initial comment on the expression for charge carrier concentration From the expression we have just given for the state density function N(k) in reciprocal space, we can directly note the carrier concentration by: n=

1

N(k)f(k)d 3k , so with f

~ fo, n ~

1

N(k)fo(k)d 3k .

(8)

b Form of the state density function in k space In order to evaluate N(k), we can use a simple cubic lattice with sides of length L. Here the conditions on the periodic limits (the Born-Von Karman conditions) are written , for example, in the x direction so that 1\J(x) = 1\J(x + L) and are applied to the Bloch wavefunctions, as in: \IJ(x) = eikxu(x) = 1\J(x + L) = eik(x+L)u(x + L) [with u(x + L) = u(x) as L is written as L = na in which a is the crystal periodicity] . This gives us the relationship e ikL = I which fixes the form of k, which has to be such that k = n (where n can be either positive or negative) . In the direction kx, one cell (the space in which we can place a state without which can having to take spin into account) therefore has the dimensions ~kx = be written in 3-dimensions, using the usual notations, as (~k)3 = 8~3 (with V = L 3).

22

22 ,

Without taking spin into account, the number of states into which we can possibly place carriers, in a unit of reciprocal space, is equal to (~~)3 = ~ . Once we introduce the spin states, which multipl y by two the preceding result, and remember that N(k) is defined for a unit volume of direct space (V = 1), then

A-4 General theory of conductivity for a regular lattice

407

we can write definitively the density function of N(k) states as: N(k) = 2 ( V3 ) 8n

V= l

I 4n 3 ·

(9)

c State density function in energy space The titled function we shall denot e as N(E) , although sometimes it is called Z(E), and it such that N(E)dE repre sents the number of states distributed in energ y space between E and E + dE, the whole still being in respect of the direct unit volume in space (V = I). Supposing that the electron effecti ve mass (m" ) is invariable whatever the value of k, and that variations in k are isotropic in the space of wavefunctions, then we can write (in approximation of the effective mass) that E =

~~: , taken with p = lik = nr'v,

E = 1m*v2 (and thus with v = ~~). In k space, as shown in Figure 2, the surface equienergy E = cte, and is a sphere of radiu s k = with radiu s

¥. In the same way the surface of the equienergy E + dE is a sphere 2

k+dk=

,J2m*(E + dE ) ( li (k + dk)2) asE+dE= . Ii 2m*

In term s of the state density function , the densit y of the states N(E)dE corresponds to the density of states spread throu ghout the k space in between the spheres of radius k (for E = constant ) and k + dk (where E + dE = constant). As the volume held between these two spheres, and shown as the greyed zone in Figure 2, is equal to d 3k = 4nk 2dk , we can write that

N(E)dE = N(k)d3k . With 3

d k =4n

(2m*E) 1/2 m*dE . - 2- ' Ii Ii

we have:

Figure 2. Representation of equi-energy spaces within reciprocal space.

(10)

408

Optoelectronics of molecules and polymers

From this we deduce that (11)

3 Expression for flux In general terms, eqn (7) describes the carrier flux and is an expression in which we can now introduce the value of N(k) given by eqn (9). Additionally, the statistical distribution function (f) is such that f(k) = fo(k) + fl (k) and with the velocities being zero at the point of thermodynamic equilibrium it is only f 1(k) which contributes to the expression for the carrier flux, which can thus be written: (7')

4 Expression for current density in reciprocal space For a force due to an applied electric field we can assume that the electric field (Ex) is applied parallel to the axis k, in reciprocal space in order to simplify the expressions used. With the force being exerted on a charge q being F(x) = qE x, eqn (6) becomes

afo f = fo- q"[Exvx as

(6')

On writing E - EF)]-I = [u(E)r 1 and posing v = (E - EF) , fo = [ 1 + exp ( ~ ~ the derivative with respect to E in the expression for fo gives :

2' .with · u' = [1 + exp(v)] , -afo = - [u(E)] -2 u' = -fOll aE 1 exp(v). = [exp(v)] " = v exp(v) = kT

As I

+ expv =

I I - fo , 1 I - fo -, soexp(v) = - - , we have II = - - fo fo kT fo

and finally :

afo fJ 1 - fo fo = - - - - = --(1 - fo) aE kT fo kT

(12)

We then find that eqn (6) gives f = fo + for-rExvx(1 - fo), in which f - fo = f1 = r-rExvxfo(1 - fo)·

A-4 General theory of conductivity for a regular lattice

409

On assuming that the field Ex is constant and uniform, the result gained from eqn (7') for the flux, now the current density (Jx) is such that:

(13) (Here the Boltzmann constant is denoted kB in order to distinguish it from the wavenumber k which defines the reciprocal lattice and is such that, for a cubic 3-dimensional system, k =

Jk~ + k~ + k~.)

III Different expressions for the current density and relaxation times and forms of conductivities and mobilities 1 Usual expression for current density in energy space Eqn (13) which gives the current density can be rewritten as a function of energy by using eqn (10), which is such that N(E)dE = 4~3 d 3k . Eqns (7) and (13) thus give : 2

Jx = q Ex

~

E

-rv2x -fo(l - fo)N(E)dE . kT

(14)

When considering both an isotropic diffusion and the approximation for the effective 2) 2 = VY 2 = vz2» (iIn W hiICh V2x = "3 V 2 = 3m*' 2E Inserting . m*v an d vx mass th en E = -2,so t h at vx this value for v~ into eqn (14) in addition to eqn (11) for N(E), and on setting A = ~~ (2m*)3 /2 we now have J x = -2 ; 2Ex 34)"[ m*kT

f

In addition, from eqn (8) we have n ~

-r(E)Efo(l - fo)AE 1/ 2dE .

fk

(14')

N(k)fo(k)d 3k, that is:

in which

(8') Introducing eqn (8') into eqn (14') results in Emax

J _ ~ nq 2Ex fo x - 3 m*kT

-r(E)E 3/ 2fo(l - fo)dE foEmax El / 2fodE .

(15)

410 Optoelectronics of molecules and polymers 2 Studies using various examples a When

is a constant i.e. T(E) = T

T

"[ can be given by the integral I of a numerator which is limited: 1= IoEmax E3/ 2foO - fo)dE. On using eqn (2): 1= -kT

r-

10

E

3

/

3 2 max 2 afo aE dE = -kT {[foE / JE

0 - 10[E

max

3 l/2fodE} . 2E

The first term in the integration by parts is zero (as fo(Emax) = 0 when Emax ---+ 00) , so that I = ~ kT IoEmaxE1/2fOdE. Moving this into eqn (5), we find the usual formula nq2"[ Jx = --Ex .

(6)

m*

b Degenerate semiconductor C'L First order calculation Here the Fermi-Dirac function goes from I to 0 in small intervals (of the order of kT) about EF. So for E < (EF - kT/ 2) we have fo ~ I , and for E > EF + kT/ 2 we have fo ~ O. On placing = "[(E)E 3/ 2, for a numerator for Jx given by eqn (5) the term B appears which is such that B = k~ IoEmax (E)foO - fo)dE. Using eqn (2) and factorising the integral gives:

B= -

l

EF- kT/2

o

afo iEF+kT/2 afo i Emax afo (E)-dE (E)-dE (E)-dE. aE EF-kT/2 aE EF+kT/2 aE

As fo ~ I = a constant between 0 and EF - kT/ 2, the first term is zero, as is also the third term as fo ~ 0 between EF + kT/2 and Emax. As between EF - kT/2 and EF + kT/2, we have (E) ~ cte = (EF) , we can write

Calculating the denominator means determining [E

10

max

E1 /2fodE ~ [EFE 1/ 2dE =

10

~E~/2 . 3

(7)

On introducing the last two results into eqn (5) we have: Jx =

nq2"[(EF) m*

Ex.

(8)

A-4 General theory of conductivity for a regular lattice ~

The more generalform

Eqn (15) can be written in the form t(E)E 3/ 2fa(1 - fa)dE

nq 2E 2 I Ia

J - - -x -

m* 3

X -

411

E max

-::....::e..------;o;------kT IaEmax EI /2fadE '

so that with eqns (12) and (17) we now have 2E J = nq x [_E- 3/ 2 x m* F

[ 00 t(E)E3/2 afo dE]

ia

.

aE

By consequence (19) with

t = _E- 3/ 2 F

00

1 a

at

t(E)E3/2~dE . aE

(20)

c Non-degenerate semiconductor Here E - EF

»

2kT, so that exp([E - EF]/kT)

1 fa = I + eXP([E _ EF]/kT) in the Boltzmann distribution. As fa « 1, we have fa(1 - fa) 2

2E

J __ ~ x -

3 m*kT

~

»

1. And

~ exp([E _1 EF]/kT) = A exp ( - k~ ) fa, and eqn (15) gives :

[ 00 t(E)E 3/ 2 exp (-~) dE [ 00 E 3/ 2 exp (-~) dE kT ia 00 kT 00

ia

1

E3/2 exp ( _

k~ )

1

dE

E' /2 exp ( -

k~) dE

With

[ 00 E3/2 exp (-~) dE

{E} = io 00

1 100

{t} =

kT

32 t(E)E / exp (

[ 00

ia

k~ )

E 1/2 exp ( -

32 E / cxp

3

= -kT

dE

-~ )

and

2

dE

(E ) , - kT

dE

we finally have : nq2

Jx = -

(t) Ex . (21) m* (We can note that the weighting factor is EN(E) , with N(E) = N(E)fa ~ E 1/ 2 exp( - k~)' and not N(E) as we might initially think) .

412

Optoelectronics of molecules and polymers

3 Expressions for mobility As a preliminary step we can note that mobilit y (IL) is classically defined by one of the two relations lvl = ILEx, or a = qnu .

a When the relaxation temperature ('t (E )) is constant i.e. 't eE) nq21

Eqn (16) written as Jx = ITi*Ex allows us to write a =

~ m* ,

=r

so that

qT IL= - · m*

(22)

b For a degenerate semiconductor

a First order calculation Eqn ( 18) written Jx = ~

nq2T(Ep) m*

. nq2T (Ep) qT(Ep) Ex, yields a = , so that IL = - - . m* m*

The more general form -3 2

;: = -Ep /

00

1 o

(23)

With 8& 10 nq 2-T T(E)E 3/ 2 -dE, eqn (19) written Jx = - - Ex, aE m*

gives nq2;: a=--

m*

q't

(24)

IL = - . m*

and

c Non-degenerate semiconductor Here , eqn (2 1) written

1

00

with

(T) =

3 2 T(E)E / exp (

roo

Jo

-~ )

dE

(E ) ,

3 2 E / exp - kT

dE

is such that a = ~ m* (T), so IL =

~

m*

(T) .

(25)

A-4 General theory of conductivity for a regular lattice

413

4 The Kubo - Greenwood expression for conductivity a General case We have already realised eqn (14) for current density, that is 2

J x = q Ex

1

2

1:V x -fo(l - fo)N(E)dE . E kT

So that conductivity is in the form (J

= q

1:Vx2 -fo(l - fo)N(E)dE . kT

(26)

j..l(E)fo(l - fo)N(E)dE ,

(27)

q1:(E)v~ kT

(28)

1 L

2

E

Eqn (26) san be rewritten to give (J

= q

with j..l(E)

=

in the Kubo-Greenwood expre ssion for conductivity initially found in general physical theory . Following on from eqn (12), we can also write (J

afo(E) dE aE '

(29)

= q kT N(E)j..l(E) .

(30)

= -

[

(JE

JE

with (JE

b Isotropic diffusions and with an approximation of the effective mass 2

2 _ V ( Vx - T

2E ) = 3m"

The introduction of v~ into eqn (14) gives (J

= q [

q~fo(l

(J

= q [

q1:~fo(l

JE 3kTm* JEmtE

- fo)N(E)dE ,

-

so with E

=

3

2kT,

- fo)N(E)dE .

(31)

We once again find eqn (27) of the Kubo-Greenwood expression in (J

but here with

= q

L

j..l(E)fo(l - fo)N(E)dE,

qt E j..l(E) = - = . mOE

(32)

414

Optoelectronics of molecules and polymers

IV Complementary comments 1 Concerning the approximation of the effective mass and isotropic diffusions a Basic expressions If the band structure is known we can express E(k) as a Taylor series, as in E(k) = E(O) + k [ -aE ] ak k=O

2E]

+ -I k 2 [a-2 2

ak

+ ... k=O

Once the band exhibits a minimum at k

= 0, laEJ ak

=0

and

m*

I a2 E(k) li2~ '

E(k)

= E(O) + li

2k2

2m*

k=O

,

with

On introducing the crystal moment, lik = m*v, we have for a parabolic band which is such that E(k) - E(O) = ~~~ (which also takes into account the origin of the

i::)

energies for electrons in E(O) = Ee (lowest conduction band), E(k) = and E = im*v2. As we have already seen, for isotropic diffusions , we therefore derive

(v~) = (v~) = (v~) = ~, so that on using the effective mass approximation we have (vi) =

3;;"

b Additional remarks concerning the approximation of the effective mass [Lun 00] When an electron moves through a device, it is subject to both a periodic potential of the crystal (Vc) and an internal or applied tension (VIA) caused by impurities due to donors, acceptors, super-lattice effects and so on. Typically, VIA varies little in space with respect to Ve . Electrons evidently see both Ve and VIA. If we assume the collisions between electrons and the lattice to be negligible , then the electronic wavefunctions must accord to

2

li2 a --2 [ - 2m ax

a\IJ + Ve + VIA ] \IJ(x, t) = ili. at

(33)

In the absence of VIA, the solutions of eqn (33) are the well known Bloch functions. The question which must now be asked is with what condition is it possible to express the solutions for the complete problem as expressed in eqn (33), and, without an evident solution, do this with the help of the known forms of the simplified problem. The response is that it is possible using carriers situated at the lowest point (k = 0)

A-4 General theoryof conductivity for a regularlattice

in a parabolic band (which exhibit energy bands with a minimum at k we can write \IJ(x, t) = F(x, t) . Uk=O, in which F(x, t) is such that 2

a + VIA ] F(x, t) = -li2- [ 2 2m* ax

415

= 0) for which

aF

ih- .

(34)

at

The function \II is thus the product of a function envelop and varies slowly due to the presence of VIA in the equation, and also a function uko(r) which is periodic and varies rapidly with the Bloch function calculated from k = 0. The advantage of this type of solution is that the complicated crystal potential effects (Vc) can be written into a single value for the effective mass, hence the name the effective mass approximation. Nevertheless, writing eqn (34) did necessitate that VIA varied sufficiently slowly with respect to Ve . In effect the potential wells associated with VIA should not be too steep implicating a low level of internal disorder and tension, and a sufficiently low frequency and tension in the applied field.

2 General laws for changes in mobility with temperature a Semiconducting system (non-degenerate) We have seen that generally speaking t is dependent on the energy of the charge carriers in 't = t (E). The classical law for the evolution of t is t = AE -s , in which s depends on the type of collision encountered. With eqn (21) which expresses (r) , following [smi 61], the calculation can give (t) ex A(kT)-s.

(35)

r,

In more precise terms, we can put t into the general formula 'teE) = 'to (k\ in which 'to is a parameter dependent on the nature of the interaction and the temperature of the lattice . When collisions do occur: •

on the lattice (collisions with phonons), when the temperature is sufficiently high, we have r = -1 /2, and 'to is proportional to T-' /2. We then have 't res = aE- 1/ 2T- 1, so that: ('t res) ex T- 3 / 2 as much as ILres ex T- 3/ 2 ;

(36)

on ionised impurities, then r = 3/2 and 'to ex T 3/ 2 , and 'tii = bE 3/ 2 , so that:

•

< r u> ex T 3/ 2 much as lL ii ex T 3/ 2 ; and on neutral impurities, then r ~ 0, 't ex TO , and

• l

-t

res

t

(37)

~ constant.

If we consider that neutral impurities are negligible, then here we also have ~ = + -L., which can also be equal to 1 = .L, + _1_ . Depending on the temperature [l

til

flre s

[l imp

regime :

-

1 ~ fl

~ ~

_1_

:ii

flre s

and IL ~ ILres ex T- 3/ 2 at higher temperatures (T 3/ 2 and IL ~ lLii ex

T 3/ 2

at lower temperatures (T 3/ 2

»

T- 3 / 2 ) ; and

« T- 3/ 2 ) .

416

Optoelectronics of molecules and polymers

b metallic conduction (cf. example given by [moo 93])

Metallic systems involve transport by electrons which are considered to be practically free and diffused by phonons and impurities, of which: -

-

the frequency of electron-phonon collisions (Wph), at a temperature higher than the Debye temperature, is proportional to the number of phonons (Nph), which is such that Nph ex T. Thus Wph = t~h ::::::: Nph ex T. With 0 ex 1:, the law of change of conductivity (Oph) as a function of temperature is Oph ex T- 1, and also Pph ex T; if the concentration of impurities (Nimp) remains independentof T, the frequency of collisions between carriers and impurities is Wimp = -t IImp ex Nimp, so that Pimp ex

_1_ = Timp

Po (constant).

If we assumethat the two collision mechanisms are independent fromone another, then we have as a total number of collisions W = Wph + Wimp, in whichthe average time betweencollisions (equallyelectron-phonon andelectron-impurity) is 1.r = .L + t ph _I_ . The upshot is that the resistivity ( p) is such that P ex 1. = .L t ,mp 't tph written, following Matthiesen's law:

P = Pph + Po = po (l

+ u'I').

Eqn (38) is useful at temperatures close to ambient.

+ _1_, and can be Tirnp (38)

A-5 General theory of conductivity in localised states: generalisation of the Kubo-Greenwood formula, thermally activated mobility, and localised and degenerate state bands

Even if the application of transport equations, usually obtained for ordered media in which it is a simple matter to define a reciprocal lattice-see Appendix A-4-is 'fanatically' classical, we shall now record how a general Kubo-Greenwood type expre ssion can be used to describe hopping transport mechanisms between localised states. We will indicate an expression for mobility and detail a new distribution function form which may be used in accordance with whether or not the Fermi level can be found in the band of localised degenerate or non-degenerate states.

I Expression for current intensity associated with hopping transport of a given charge placed on a specific energy level 1 Transcribing transport phenomena into equations We will consider that the transport here is of a given charge (q) which is initially placed on a certain energy level (E) (occupied with the probability f'(E) = fo which, very generally speaking, is the Fermi-Dirac function) . The tran sport is through hopping over a potential barrier of height U and situated between two sites A and B, each pair of sites for the moment having a given energy E. In the absence of an applied electric field, as in Figure l-a, the probabilities of occupation (fo) are identical for the two sites. Per unit time , transition probabilities from A to B and from B to A above the potential barrier U are also equivalent. We can therefore write PAB = Po exp (- ~) = PBA in which Po is the transition probability per second from A to B or from B to A with zero barrier and can be written as Po = ~ = ~ (as is the frequenc y of oscillation between two sites unaffected by barrier or external field). Once a barrier is set up, but still there is no external field, we have therefore

418

Optoelectronics of molecules and polymers d

(a)

d

(b)

Potential energy I

x

Figure 1. (a) A and B sites separated by a barrier over which hop carriers ; and (b) the modification of potential energies following application in Ox of an electric field (Ex).

or rather U

t

= toe IT .

(1)

In the presence of an applied electric field Ex which is directed with respect to Ox, as shown in Figure l-b, the potential energy W = qV for a charge q is reduced at Band increased at A (when q > 0). With Ex = -dV /dx , and supposing that Ex is uniform with respect to Ox, on taking the origin of the potentials in terms of O(V (0) 0) we have:

=

YeA) - YeO)

1 = 10 = 1 = ~O =

A

dV

o

Exdx

= Ex(O -

XA)

Exdx

= Ex(O -

XB)

A

2

B

V(B) - YeO)

o

dV

= Ex-d = YeA) . = -Ex-d = V(B).

B

2

For a charge q situated at A or B, the modification in energy due to the application of Ex is, respectively, W(A) = qE xd /2 or WeB) = -qExd/2 . It is noticeable that the energy increased by (qE xd/2) at A and decreased by the same quantity at B. Here we finally have an asymmetry with the A wells increased in height and the B wells pulled down . At A the barrier to overcome is reduced and becomes: V A = V - qE xd /2. At B the barrier is increased and can be written as VB = U + qE xd/2 . Consequently, in the presence of an electric field the transition probabilities, P AB and PBA, are no longer equivalent and become:

PAB PBA

= Po exp (

U - qExd/2) kT

= Po exp ( -

V

Exd

I +9 / 2 = -e kT t

+ qExd/2) = -e I _9 Erf 2 kT

t

1[

~ -

t

~

qExd/2] 1 + --kT

I [ qE xd/2] - I - --r kT

A-5 General theory of conductivity in localised states

419

2 Calculating the current intensity due to hopping mechanisms The current due to transport of a charge q between A and B, Ie, is the resultant of charge transport from A to B and from B to A, i.e. Ie = IA--+B + IB--+A ' It is the total charge moving per second in both these directions. If fo is the probability of site occupation at equilibrium and if 0 - fo) is, inversely, the probability of non-occupation, the quantity of charge moving from A to B can be written as IIA--+B I = qfoO - fO)PAB . Similarly, IIB->AI = qfoO - fO)PBA . As the currents lAB and IBA are in opposing directions, the resultant can be obtained:

which gives: Ie

= qfoO -

I

qExd j2 ]

[QEXf2

fo)~ e+

k

-

e----rr

xd /2) I sinh (qE = qfoO - fo)-2 -T kT

~

I [qExd] qfoO - 1'0)-2 -- , 2kT T

or rather:

I [qExd] le~foO-fo)T

kT

.

(2)

II Expression for current density and thermally activated mobility 1 Expression for current density relative to transport at a particular energy level The state density function for an energy level E being N (E) (number of states per unit volume of material) with in the volume V of a unit section is V = [S x d](S=I) = d, and the number of states with energy E is thus N (E)d . This is the number of individual charges (q) situated at sites with energy E (with T = T(E) for carriers leaving the sites). If we take a unit section between two sites A and B separated by the distance d, the current passing through this section thus becomes a current density with the value:

I(E)

2q = leN(E)d = -foO T(E)

xd /2) fo) sinh (qE -N(E)d kT

d] [qEx q ~ T(E) foO - fo) kT N(E)d, which yields :

d2 1 ICE) ~ q2Ex--foO - fo)N(E). 1:(E) kT

(3)

420

Optoelectronics of molecules and polymers

Figure 2. Schematisation of all possible hops through the section S = 1 giving rise to the currentJ(E).

2 Generalisation of the form of Kubo-Greenwood conductivity In order to assimilate all the possible hops-shown in Figure 2-through the section S, the length d must be placed in to terms of the mean free pathway (.C) the mean distance between two hops. In addition the current density, which involves all carriers from a volume V traversing the section S in both senses whatever their initial energy (E), can be written as:

J = q2E x

~

£2 I - - f o ( 1 - fo)N(E)dE E teE) kT

(4)

Again using the definition of conductivity (0') as that given by the relationship

J = crE x can be used to give a Kubo-Greenwood type equation : 0' = q

with

f

I1-(E)fo(1 - fo)N(E)dE,

£2 I !-L(E) = qt(E) kT ·

(5)

(6)

Comment With L = vxt(E), and with the help of eqn (4), we can go directly to: (J

=q

qv~t(E)

1 E

kT

fo(1 - fo)N(E)dE ,

a form of (J deduced from the Boltmann transport equation shown as eqn (14) in Appendix A-4. We can thus be content to use eqn (14), Appendix A-4, in which E = vxt(E) is introduced and from which we can directly obtain eqn (4). However, such a path tends to hide the physical reality of the problem which we have detailed in this Appendix .

3 Thermally activated mobility Following on from eqn (1), we have eqn (6) gives: 11-

t

= 'to exp

(&), and

on placing 11-0 =

= 11-0 exp ( - ~ )

This equation shows us that the mobility is thermally activated.

~~~ , (7)

A-5 General theory of conductivity in localised states 421 Comment

From eqn (6), we can write that

On posing

£2

D = -,

(8)

t

which has as dimensions a diffusion constant, we find Einstein's relation: kT q

D = -~.

(9)

While the study of transport with respect to the direction of the electric field does not a priori require the introduction of the coefficient 1/6 , we can have the isotropic diffusion . I (0) with P = - . r

III Approximations for localised and degenerate states Here the calculationsfor conductivity can be performedusing the Kubo-Greenwood relationship i.e. (J = q f ~(E)fo( I - fo)N(E)dE, but require,in general, thefollowing approximations: -

as thestatesare neighbours at the levelE = Ep, we assumethatN(E) ~ constant = N(Ep); and ~ = ~o exp (- k~) = constant with respect to E, as in ~ (E) = ~ (Ep).

The calculation for (J requires therefore the evaluation of (J

= q

~o exp ( - ~) N(Ep)

f

fo(l - fo)dE.

We must therefore go on to evaluate f foO - fo)dE. This can be done using eqn (2) of AppendixA-4, so foO - fo) = -kTt . Figure 3-a represents the Fermi function and Figure 3-b the function - t . At not too high a value ofT, (T ~ T, = 300 K), the above schematicrepresentation shows that t behavessimilarly to a delta function, so that

Thus:

f f

kT

(~~) dE ~ kT.

foO - fo)dE ~ kT.

When a high number of defaults result in deep localised states within the gap, we find that (Be - Ep) » kT, and carriers cannot be thermally excited into

422

Optoelectronics of molecules and polymers (a)

dfO dE

(b)

/~...TTo=OK

•.

. . . . . I \ . . .( '"/ •.•••••

.' I

1

r,:» T 1

X/ ••••••

::

.

E

Figure 3. Representations of: (a) the Fermi function; and (b) the function ~ .

the conduction band. These carriers remain localised around EF in a degenerate band of states. The transport process in the neighbourhood of EF is realised by bringing into playa density of charge carriers given by n = N(E)foO - fo)dE ~ N(EF) foO - fo)dE ~ kTN(EF)· The conductivity is thus in the form o = q kTN(EF)f.LO exp (and is thermally activated by the form of the mobility.

f

f

Ji.)

A-6

Expressions for thermoelectric power in solids: conducting polymers

I Definition and reasons for use 1 Definition Under the effect of a temperature gradient, there is a diffusion of carriers from hot to cold points which is associated either with a concentration gradient as in the case of semiconductors, or a velocity gradient of carriers in the case of metals . The result is an excess charge and, in effect, the birth of an internal electric field (E). The resulting ~

~

-----+

current is the sum of two parts : j = crE + M gradT

At equilibrium the current is zero and the internal field is such that: ~ ~ gradY VA - VB E = S gradT, so S = - - - = - - - gradT TA - TB

!1V !1T

in which , by definition, S is the Seebeck coefficient.

2 Reasons for use The parameter, called thermoelectric power (TEP) is: •

•

an intrinsic property of a material and is independent of the geometrical dimensions and insensitive to faults such as cracks, in contrast to other properties such as conductivity; and essential to distinguish metals from insulators and understand which carrier is in the majority in a particular material.

424 Optoelectronics of molecules and polymers

II TEP of metals (EF within a band of delocalised states) A rigorous calculation must be performed within the context of the Boltzmann transport equation:

For a metal close to the Fermi level, we have (d In o I dE) < 0 and therefore Sm > O.

N(E)

On developing the calculation with o(E) ex aE' and [dlncr(E)/dElEF ~ liEF) we have the following equation:

Sm

~

k Jt2 kT

-- q 3 EF

~

several u V/K .

°

Thus Sm has a low positive value and Sm = f'(T) is linear with Sm -+ with temperature. Qualitatively speaking, we can say that with a good metal the thermal gradient does not practically 'destroy' the equipotential at its surface and that ~ V ~ 0, so S is very small .

III TEP of semiconductors (SC) (EF in the gap)

A-6 Expressions for thermoelectric power in solids: conductingpolymers 425

1 Preliminary remark In a metal the diffusion of hot electrons of charge -q, takes place above EF and therefore the effective negative mass is such that the associated force is directly opposite to the diffusion velocity: :::} VA < and VA - VB < 0, which gives Sm > 0, as we have already seen . For SCs , the carriers (-q) have m" > 0, so, only in p-type SCs , we have carriers with +q and m" > 0. In order to determine the force applied on the carriers, (+q or -q), we can use f = m*y with y IIv and m" > or <0).

°

°

A

2 An ideal n-type semiconductor Locally around A, the concentration of carries increases with the effect of t::. T (n ex T 3/ 2 exp(- [Ec - EF]/kT) . The result is that electrons diffuse from A towards B (m: > fel Iv) where a charge < 0 appears. And leaves a charge > 0 at A. We can thus write that VA - VB > and TA - TB > 0, so that SN < 0 .

°:: }

°

r. More quantitatively, without going through the rigorous treatment of the transport equation) we have S = (Kelvin's 1st relationship), where IT is Pelletier's coefficient. For electrons, with the energy transported per unit charge and by taking the potential origin as EF, and then with

¥

Energy = Ppotential + Ekinetic, Epotential = Ec - EF, Ekinetic

~

Ec - EF +2kT 2kT : IT = - - - - - -

-q

C

Ec

E"

---f 2kT

426

Optoelectronics of molecules and polymers

Finally: with here B = 2 Thus SN < 0 and typically goes in value from a hundred or so I-L VK- 1 to more than several mV K- 1 .

3 An ideal n-type semiconductor The diffusion of holes (+q , m~ > 0) with ftl [v results in charging B positively (while leaving a negative charge at A). So we have VA - VB < 0 and TA - TB > 0, and that Sp > O. Thus :

Sp = -k [EF - s, +B ] , q kT

here with B = 2

A f

T+~T

4 Comment on amorphous semi-conductors for which Ec - EF = b.E - aT. For a n-type semiconductor, for example, SN = -

~q [b.E - ~ + B] kT k

while S = f(l/T) and In a = f(l /T) exhibitthe same slope . The ordinate at the origin for S = f(l /T) yields a .

5 A non-ideal amorphous semiconductor with EF below its states in the band tails Here the carriers have a thermally activated mobility in the band tails and I-L = I-LO exp( - U I kT ). The form of the tail band is such that N (E) =

N (Ed (E - EA)S (Ec - EA)S

and calculating S within Boltzmann restrictions gives S=

_~ q

kT

[EA - EF kT

(a ~ aoC s ( Ec _ EA

+ ~] Cs+1

)S (EA - EF + exp kT

U)

A-6 Expressions for thermoelectric powerin solids: conducting polymers

427

as discussed in Chapter V, Section IV-I) .

Conclusion The slope of In (J = f (1 IT) is larger than S = f (1 IT ) by an amount U, and is characteristic of this transport mechanism within tail band localised states (in the absence of localised states in the band tails, the two slopes are identical.

IV TEP under a polaronic regime We will distinguish between high and intermediate temperatures in a manner similar to that used for conductivity.

1 High temperature regime Here, S can also be deduced using Kelvin's l st relationship, S = rI/T in which Pelletier's coefficient (energy transported per unit charge) is such that we can suppose that the polaron does not transport the vibrational energy IT = ~ (Eo is the energy which characterises the thermal generation of carriers and Eo = Ep - EF). From which we can deduce that in the high temperature regime, S = -

~

(f¥).

2 Intermediate temperature regime The regime implicates a VRH transport mechanism and, as we shall see in the following Section, S follows a law of the form SIT = f(T- 1/ 4 ) .

3 Other regimes It has been found that for amorphous semiconductors, S ~ Cte with respect to T. Nagels proposed that with the concentration of polaron s being constant with T, S follows Heikes law which is of the form S ~ -(k/q)Ln([N - nj /n), in which N is the number of sites and n is the number of carriers .

V The TEP for a high density of localised states around EF 1 Initial hypothesis If EF were to 'dive' into a band with a high density of (localised) states, it has been supposed that the expression used for a metallic system for the TEP would still be

428

Optoelectronics of molecules and polymers

valid by Cutler and Mott. In reality, however, experimental results have shown that S i= 0 as T ~ 0 in complete discord with the hypothesis.

~

BC

1----------

BV

...

In fact Nagels has shown [Nag 85] that in the low temperature domain we can write: S= _

Jt2

~KT {d(lner(E)}

3 q

dE

E=EF

and that c(E) is expressed as a function of N (E) for which we assume that close to EF there is a linear variation such that N(E) = N(Ep)[1 + y(E - Ep)] with yW « 1, in which W is the height of the hop .

2 The result in VRH Here we follow the work of Nagels in J. Non Cryst Sol. (1983) and consider the VRH condition with respect to the optimisation of both the hopping distance and the hopping height (W(E», the calculation of S, with y = (1/N(EF)[dN(E)/dE]E=EF, gives:

Comment The advantage of the above calculation with respect to other possible routes is that it brings in the factor y which takes into account variations in N(E) near to EF and does this independently of the origin of the states in the gap disorder, polarons etc.).

A-6 Expressions for thermoelectric power in solids: conducting polymers 429

VI General representation In (J

E

(E)

VRH .---

I

;

Ec

i

EA - E.,

!

:.1-

_

q

Iff

Iff

VII Real behaviour 1 Generallaws a Different levels of conduction participate in parallel The law is derived from the general equation itself generated from the Boltzmann formalisation, and is such that

As an example, we can site the conduction by both electrons and holes in S = GeS e + GhSh , Ge + Gh and that in an intrinsic semiconductor we have

b Differing domains in series Kaiser proposed [Kai 89] that during a drop in temperature, is remain proportional to the thermal resistance (Wj) of each domain, sothatfromS = -!1Vj!1T = -(!1Vl +

430

Optoelectronics of molecules and polymers

fj, V2) 1 fj, T = (fj, Ti l fj, T)SI + (fj, T21 fj, T)S2 we can deduce that S (W2/W)S2, with W being the total resistance.

= (WI /W)SI +

2 Behaviour as a function of dopinglevels

a Highlydoped conducting polymers In addition to the metallic component in S, it has been observed that at low temperatures there is a distortion of which Kaiser brought into account by using a non-linear component A. *(T)T associated with effects on the carriers due to phonons. The resulting equation is therefore:

b Moderately dopedconducting polymers Here there are two ways of looking at the scenario:

•

•

EF is more or less outside of the extended states and the system tends towards a semiconductor character => SIT ex -y(ToIT) X(see also Section y) or the system is an ideal semiconductor in which case exhibits a slope [Ec - EF] /q with l iT) ; and that we have a heterogeneous structure (Kaiser, Phys Rev 1989), with conducting fibres separated by barriers and we have S = (W t/W)SI + (W2/W)S2 = XT + CT 1/ 2 . The first term in this equation characterises the (metallic) transport along the fibres while the second term, for inter-barrier hops, follows a VRH type law with respect to T I / 2 de type proposed by Mott. If the barriers are thin and conduct heat better than electricity, we can simplify things by assuming WI » W2 and that S ~ XT (see also Section a).

c Poorly doped conducting polymers (EF is placed within a quite large gap that has few states and the material is close in character to an insulator). Whether looking at polaronic states or states tied to locali sed defaults, at low and intermediate temperatures, in principle they both follow a VRH law in the form SIT ex f[T- 1/ 4 ] . Indeed, under certain conditions, such as in the impurity band, Heikes law can be envisaged so that S ~ -(k /q) In([N - nj /n) is reasonably constant or decreases only slightly with T.

A-6 Expressions for thermoelectric power in solids: conducting polymers

431

3 Representational graph

lsi Mode rate to weak dopin

Weak doping:

(SC'mlronducdng statr "Ith ffl' loaal l....t . ta les) Is I • Iklq)( (MAT) + HI

s .. etc with T

II rik". la",'

Weak doping: (ma o) IQClIIi....t

l~l <'l )

Is I rr = 1 f(T .IJ~) trong dopi ng: S > 0 and S =XT (' meta l')

_ .J.*: distorsi o ".

T(K)

4 An example result An useful example is that with reversible doping-performed with ion implantation at 30 keVin poly(para-phenylene)-followed with respect to the changing positive and negative value of S [Mol 98]. In the Figure below, the curve (a) shows the variations of a sample initially made p-type using iodine and then doped to n-type by an increasing flux of n-type alkaline ions. Curve (b) shows the effects due to a sample undergoing initial n-type doping with alkaline ions and then a subsequent implantation with an increa sing flux of halogen ions.

60 Initial iodine implan tatio n

~

40

;;. :::l.

.

~ ~

D: Flux (ions cm"2)

20

OJ

~ 0

e,

..E'" 0

:S ~

=

(D 5 X IOU). followed bv C. impla nta tio n

0

- 20

- 40

- 60

A-7

Stages leading to emission and injection laws at interfaces

I Thermoelectric emission and the Dushman - Richardson law With W s » kT , we have E - EF » kT and the Ferm i function tend s towards the Boltzmann function: f eE)

~ exp( -[E -

EFl/kT) = exp ([EF _

h;~2 ] / kT) .

So that an electron is emitted in Oz-simpl y through therm al agi tation- and without takin g into acco unt a reflection coeffici ent, the equ ation [(E)selon 01. - EF1:::: Ws must be true , as schematised in Figure I . Given that the energy is uniquely kinetic than the emission co ndition is thus [El in 01. =

h2k1.2

2m :::: EF + Ws =

h2k 2

2~O .

(I )

We will define the numb er of electrons as dn in the reci procal space volume d 3k . Th e latter is divided into elementary cells so that ~ kx ~ ky ~k1. = ~k 3 and

~k3 = 8~3 . On reasonin g that V = I (dn will be a density) and the possibility of placin g two electrons per ele mentary cell (due to electron spin), we have: 3k

dn

= 2 x [number of cells in d 3kl x f(E [k]) = 2-d

8n 3

f'(E) .

Th e elementary current density (dl 1. ) emitted in the direction Oz and effective for dn electrons is therefore h 2 k1. d 3k hk 1. dJ 1. = d p V1. = e dn V1. = e dn --;- = e 4n 3 exp (EF/ k'T) exp( - E/kT) ~. The emi tte~ current density ca n be obtained by adding over all electrons which have a vector k with a component k1. which complies with the already establi shed condition eqn (I): h2 k 2 h2 k 2 __ 1. > E +W = ~ 2m - F s 2m ' whatever the other compo nents k, and kyo

434

Optoelectronicsof moleculesand polymers k,

w, EF Conduction band Ec

--I:..::..I:..::..:..::..:..::..:..::I-----'--....I.--1~-___jf+--_+----=--...>I.<:'--t_-_+f__.

Metal

Figure 1. Schematisation of the thermoelectric emission condition.

Therefore:

With kzo = 2m(E~tWs), on calculating by way of the well known results for integrals

1

+ 00

-00

e- ax2 dx = ~ and a

+ 00

1 zo

I 2 ze- a z dz = _e- a zo , 2a

we arrive at the desired equation:

(2) Exactly speaking, a reflection coefficient R should be introduced to take into account any possible reflection during emission. An example of this calculation can be given with tung sten (Ws = 4 .5 eV). 1.2 x 106 A m- 2 K and ~ ~ 19 at T 2700 K, the calculation gives With A J = 2.2Acm- 2 .

=

=

II Schottky injection (field effect emissions) Here we have an increase in the thermoelectronic emission due to the lowering of the barrier (Ws in thermoelectronic emission) due to the electric field effect applied at the metal (electrode) interface.

A-7 Stages leading to emission and injection laws at interfaces

435

1 The potential barrier at the atomic scale On the atomic scale at a metal-vacuum or dielectric interface , the potential barrier observed by an electron extracted from the metal is not an abrupt step (see Figure 2-a) . In effect, an electron situated at Z > 0 in an insulator is not completely free and to distance itself from the metal is must overcome the attractive force exerted by the latter material due to the image charge (+q) which the electron induces in the metallic electrode (see Figure 2-b). For an electron (-q) on the z abscissa, its image (hole +q) is at a distance 2z. The force which is therefore exerted on the electron, which is in an insulator that exhibits absolute permittivity s, is thus

_

q2

F= -

4TI£(2z)2

q2

~

= - - -2 = -grad We

16mz

'

in which We is the electro static energy. In l -dimension, We(z) - We(oo) = We(z) = -

z lz l Fdz =

00

00

q2 q2 = --16TI£z 16TI£z

--2

(with the origin s of energy at infinity).

Comment The above representation is only acceptable when the distance z is large with respect to the interatomic distances . When z ~ 0, We(x) is brought towards the potential energy level at the bottom of the conduction band (Ec) . When z = 0, we join the energy of the electrons to Ec . 2 Emission conditions: Schottky emission law and the decrease in the potential barrier by field effect a Without an applied field Here the condition (E)in Oz ::: EF + Ws must be fulfilled (in the Oz axis) using the metal side as the origin of potentials in the lowest conduction band (in Ec ) . With the Potential energy

o v W, W(z)

=. 167lEZ

r

+q Metal

z

q2

vacuum

'4

~.-q

(elef tron) Zz

.'

Figure 2. Graphical representation s of: (a) an electrode potential as seen by an electron at the atomic scale; and (b) the charge image (+q) of an electron.

436

Optoelectronics of molecules and polymers

energy of electrons exiting being solely kinetic energy, we should be able to apply condition (1) which can also be written as tmv~ ~ EF + W s.

b In the presence of an electric field The field is applied so that the metal has a negative potential (cathode) and the part external to the metal is positive . We have -

F

----+ = -grad We = -qEa ,

and We(x)

=-

10t

qEadz

= -qEaz.

Thus, the total potential energy of an electron situated at a distance z from the surface becomes:

q2 WTotal= - qEaz - - - .

16€z

The resulting curve presents a maximum in Zm , which is such that

(

= 0,

OWTotal )

OZ

and Zm

=(

6q

) 1/2

1 Jt€Ea

Z=Zm

The result is that

The work function is reduced by an amount b. W s = IWTotal(zm)l, a value not more than

W~ =

Ws

-

qlfi (see Figure 3).

The emission condition (I) now becomes the condition (1/): tmv~ ~ EF + W~.

Potentialenergy

o

--

w,

--- ---- --------

Dielectric or vacuum Metal

q2

WTotal = -qEaz - -16

nez

Figure 3. Energy diagram describing effects due to application of an electric field.

A-7 Stages leading to emission and injection laws at interfaces

437

Log JSch Schottky straight line //~

Log J//

.... ~

/ Effect due to space charge

Figure 4. Representation of Log JSch = f

(JE;;").

Eqn (2) for current density is this still usable, as long as W s is replaced by W~. in teh presence of an electric field the emitted current density thus becomes 2

J = JSch = AT exp

-

( [

rqE;])

ws - kTqy' 4rr'E

.

(3)

For a give value of T, Log JSch = f(JE;;") is a straight line. Nevertheless, in practical terms, we can observe at low values of E a deviation from the linear law and this is due to the appearance of space charge close to the emitting surface (see Figure 4) .

b Comment using numerical example With E, = 104 V m -I , we have (Figure 3) Zm ~ 0.2 IJ-m, while with Ea = 1019 V m -I, we find Zm ~ 0.7 nm. The height of the potential barrier is reduced in the first case by ~ Ws = 4 x 10-3 eV, and by ~ Ws = 1.2 eV in the second. In a strong field, ~ W S ~ W S and the height of the barrier to overcome is extremely low, so that breakdowns can occur under high fields (Ws[Cs] = 1.8 eV; Ws[Th] = 3.35 eV and Ws[W] = 4.5 eV).

III Injection through tunnelling effect and the

Fowler-Nordheim equation 1 The problem As the applied electric field increases, the height of the barrier observed by electrons close to the Fermi level actually decreases. This in tum greatly facilitates the passage of electrons over the barrier (field emission due to a Schottky effect). In fact, another mechanism appears once the field is increased. As shown in the qualitative example just above, as E, increases Zm decreases; also we can perceive a qualitative decrease in the barrier. If the barrier decreases sufficiently then the probability of electron presence, initially close to that of the barrier (in terms of Z is equal to 0), turns to being non-zero at the other side of the barrier. This is due to the wave like character

438

Optoelectronics of molecules and polymers

of electrons , which is accounted for in the probability of presence at the point z being written as P(z) = 1'\jJ(z) 12. The barrier transparency can be calculated for simple cases . Before going in to the calculations, we can remark that if E, increases , then the effect due to the force image-which is predominant in a weak field and in Schottky emissions (close to z = zm)-diminishes when we are at a sufficiently high value of z. This can be a point, for example, in the region of z ~ ZF (defined in Figure 5 in which ZF corresponds to the point at which E = EF). On the side of the insulator, the potential energy there is simply approximated to using the field effect of the electric field so W(z) = -q E, ZF, in which ZF is the size L (such that ZF = L) of the barrier seen by electrons situated at E = EF. Thus q E, L = Ws, that is Ws L=-

(4)

qEa

In effect, the higher the value of Ea , the more narrow the barrier. In the case of tungsten, we can use Ws = 4.5 eV and E, = 4.5 x 109 V m- I to calculate that L = 1 nm. At high enough field strengths, the barrier becomes sufficiently narrow to present a non-negligible transparency to electrons free in the metallic electrode.

2 Form of the transparency (T) of a triangular barrier The transparency

cn of a barrier is usually defined by: Flux of transmitted particles Flux of incident particles '

T=----------''----

in which the flux represents the number of particles which traverse an unit surface area per unit time. The flux can this be represented by the product of the probability of

o

__ ___ ___ ____L zr =L

I I I I I I I I I

Ws

Function (E) for T ,*OK forT = OK .... .................. ......... ...

--. 1

"""1---

Line of the equation W=-qE.z

.......::::w

(E)

Figure 5. Diagram of potential energies for tunnel effect emissions.

A-7 Stages leading to emission and injection laws at interfaces

439

presenceper volume with a unit section and a length v (if v denotes particle velocity) (see Figure 6). It is possible to write eqn (5) for a barrier of length L by assumingthat the initial velocityof free electronsin the metal (Vi) and their velocityafter traversing the barrier (vd are such that Vi ::::::0 VI: 2 T _ vtltlrtransmisl - viltlrincidentl 2

::::::0

2 2 Itlrtl _ ItIr(L)1 Itlril 2 - ItIr (0) 12 .

(5)

In this problem, it is reasonable to think that the most important part of the injected current, following the applicationof a field, will come from electrons which have an energy close to the Fermi level as the distribution function N(E) has as maximum E ::::::0 EF . Thus the expressionfor T of the barrier is given by eqn (5) in which L is the size of the barrier observed by electrons for E = EF (Figure 5). In order to calculate the size of the barrier more precisely, we shall have to look at the origin of the energy at the EF level by treating a classical rectangular barrier with quantum mechanics (see Figure 7). Weshalllook at electronswith an energyE,just aboveEF anddenotetheir velocity v when z < 0 and v' when the electrons are at z > o. Generally, the conservation of energy for an electron placed initially at the level E, (neighbour to EF) and then at A after having traversed the barrier can be written, using the notations detailed in Figure 7, as: E, =

2I m v2 = 2I m v12 + U(z)

from which can be deduced m2v'2 = p'2 = 2m(Ei - U(z)), so that with k'2 = ~~ we have k'2 = 2m(\~U(z)) . For E, < U(z), k' (z) is imaginaryand I

.

k (z) = J

[2m(U(Z)-Ei)] 1/2 n}

The wavefunction, in the form ei kz for free electrons and with a real k, takes on the form tIr(z) = eiftk'(z)dz. .

T can be wntten therefore as T ex

:..

ItIr(L) 12 -2 L ( 2m ) 1/2 [U(z)-Eil l / 2d z 2 = e fo fiI , ItIr(O) I

~:

z = (v ~tht=l=

V

Figure 6. Geometric parameters required to define flux.

440

Optoelectronics of molecules and polymers

Figure 7. Representation of a triangular barrier.

and definitively

T

= feE, W s) exp

2m] [ ! -2!if

Jo[L [U(z) -

1/2

Ed

l/ 2

dz

I

(6)

.

3 The Fowler-Nordheim equation

In the relationship (6) for T, the term to integrate, [U(z) - Ed , is such that: [U(z) Ed

= Ws - qEaz - Ei Thus

l

L

o

= u(z), so that u'(z) = -qE.

[U(z) - Ed l/2 dx =

lL

[u( z)]1/2 dz = - I

l

L

[u(z)]1 /2du(z)

qE 0

0

I2 ( u(z) = -qE 3

3/2)L , 0

and on taking into account eqn (4):

1 L

1 2

[U(z) - Ed I dx

2![

= - qEI 3" 2 I

W s - qE

~ __ W~/2 , as

3 qE

Ws - Ej ]3 a qEa

/2

3 2

- [Ws-Ed I

I

E, «Ws

(and notably at T = 0 K). Finally:

T=

2m) 3"2qEI W~/2 ] .

rrs. Ws) exp [- 2 ( li 2

1/2

(7)

a

As L = ~:' we can see that (T ~ 0 K) the transmission coefficient is an exponential function of the size of the barrier. Only electrons very close to EF can leave with a non-negligible probability.

A-7 Stages leading to emission and injection laws at interfaces

441

When T =1= 0, electrons situatedjust above EF are privileged as even though they are few in number the actual barrier they have to cross is thinner. The Fowler-Nordheim calculation performed at OK does nevertheless take into accountall electrons.In using the fact that the currentdensitycontributionfrom levels situated in the interval [Ej, Ej + dEj] is in the form dJ = q N(Ej)T(Ej)dEj, the sum over all the E, states results in a current density of the form: J=

q3E~ exp[_~(2m)1/2W~/2] .

8JthWs

3!i2

qEa

(Note that N(Ei) dEi is the number of electrons hitting the unit surface area per unit time with an energy between E, and E, + dEj). Numericallyspeaking we can use this equation as in: 3 2) a exp ( -6.83 x 109 _ Ws /_ J(Am- 2) = 1.54 x 1O-IO_E2 Ws Ea

.

The above relation shows that the current density emitted for this process is negligibleat E, < 108 V m- I , and that it increases rapidly at higher field strengths. = f(J) arethe linesdrawnfor thesetypesofemissions Ln( ~2) = f( ~) or Ln( a and linear behaviour over a decade of l/V can permit determination of W;

b)

Comment The obtainedequationsrelate to emissions into vacuum (barrier Ws seen by electronscomingfrom the metal).When an emissionis into an insulatorthe barrier to overcomefor the electronscomingfrom the metal isjust WB = (Ws - 'tJ, where X. is the electronaffinity of the insulatoror semiconductor. WB can be substitutedfor Ws is the equations (see the Fowler-Nordheim equation is Section IlIA of ChapterVI).

A-8

Energy levels and permitted transitions (and selection rules) in isolated atoms

This Appendix gives a brief, but ultimately essential description of the construction of energy levels in an atom which appear under the effects of succe ssive perturbations, going from the most to the least intense .

I Spherical atoms with an external electron Typically the titled scenario resembles that of hydrogen atom s, but it can also be applied as a first approximation to the more general problem of describing an atom with more than one electron (through the potential, observed by an electron, being approximated as a symmetrical sphere).

1 Energy levels and electron configuration [Cag 71] Given the symmetry of the system under study, the energy levels can be obtained by assuming that each electron moves, to a first approximation, within a potential sphere which originates from the nucleus potential and the spherical potential which globall y repre sents the effects due to all other electrons. The states of the electron orbital s (without therefore taking into account spin) are written using an electron orbital wavefunction (1/Jn,l.m/ = Rn,18t,m\
+

-1,+1'

444

Optoelectronics of molecules and polymers

In addition, the invariability of the Hamiltonian of the system during a rotation shows that the energy levels (E) of a spherical system depends only on the two quantum numbers nand 1so that E = E(n , 1). Thi s can mean that the system is highly degenerate and at a given energy level E(n, 1), we can associate [21 + 1][2s + 1] electronic state s repre sented through electronic wavefunctions ('!J(n, I, mi, m.) 0 , using Dirac notation, In, 1, mj , ms from orbital and spin wavefunctions. In order to know the total energy of an atom , we need to sum over all the energies relating to each of its electrons. Thi s requires an understanding of the spread of electrons with respect to layers and sub-layers, as defined by the quantum number nand 1, respectively. In other word s, the atomic configuration of the atom must be known. To denote this, each layer is symbolised by a number equal to the principal quantum number (n) following by a letter which repre sents the value of the quantum number I (for 1 = 0, 1,2,3 ... the letters are, respectively, s, p, d, f ...). When several electrons are in the same sub-layer, we just write the symbol denoting the sub-layer once , and then add an indice indicating the number of electrons in that sub-layer. As an example, we can write the configuration of sodium, which has II electrons in all, as 1s2 2s 2 2 p6 3s. The filling up of layers with electrons occurs in a specific order defined by Klechkowski's rule, which states that it follows the increasing order (n + 1). For a value given which is the same as (n + 1), then it is the value which corresponds to the lowest value of n which is filled first, as shown in Figure 1.

»

2 Selection rules In this Section we finally have to con sider the conservation of kinetic momentum and how, for a lone atom with one electron, there is a radiative transition . this occurs when the variations in quantum numbers of the energy levels involved concord with : • •

~n = any integer (as the principal quantum number is not tied to kinetic moment and governs only the energy level) ; and ~l = 1 (as in, for example, the transition from the p-state [l = 1] to the s-state [I = 0] is radiative, as for the transition d -+ p, however, the transition d -+ s is forbidden and is therefore impos sible with photon emission.

II n

I

2

3

4

5

S

S

Ws

1 !-4 0 I 2

S-

I S

rI

"p " :tV If; IitP ;y

cV

d

Figure 1. Filling up layers following Klechkowski's rule.

A-8 Energy levels and permitted transitions (and selection rules) in isolated atoms 445

II An atom with more than one peripheral electron In addition the electron configuration (y) , which uses only the n and I quantum numbers to repre sent energy levels and only includes a global, symmetrical, spherical potential repre sentation of electro n interaction s, we need to take into account success ively: •

•

the exact electrostatic interactions between electr ons which introduce into the Hamiltonian of the system a perturbation operat or Hee, which raises the degeneration of the system; and weaker perturbation terms which progressively raise the remaining degeneration in the energy levels. The nature of the perturbation terms which must be succe ssively introduced depend s on the type of atom under study-whether it be heavy or light , with the limit generall y around the mercury atom (Z = 80) , along with its environment, for example whether or not it has been inserted into a matrix. The latter effect has been discussed with respect to atoms with incomplete inner layers in Chapter VII, Section II.

1 First effect produced from the perturbation H ee due to exact electronic interactions The Hamiltoni an of the system is only invariant when there is a simultaneous rotation of all electrons. Thi s can be assoc iated with an orbit al resultin g kinetic momentum operator and a spin resultin g kinetic momentum operator. We can show, relatively simply [Cag 71, Mol 81], that the energy levels from an electron configuration (y), under the influence of a perturbation Hee , correspond to energy levels (E) in the form E = E(y, L , S). For an atom with two periph eral elect rons, respectively characterised by quantum numbers I I and s I for one, and band S2 for the other, L and S are such that they vary by whole values in the Clebsch-Gordan series, so that L = II - lz II + 12 and S = S ] - S2 S l + S2. The electron states are, for their part, written using a wavefunction in the form Iy, L, S, ML, Ms ), in which ML = -L, (-L + I), . . . . + L and Ms = -S , (-S + I), .. . , +S, so that the energy levels are degenerated (2L + I) (2S + 1) time s. The fundamental state is thus determined from Hund 's rule which indicates that the lowest energy terms that for which: •

•

S is the maximum (because from Pauli 's principle, taking on the maximum spin implicate s a distinction between states by their orbital states which must thus b different and minimi ses electrostatic interactions by distancing electrons); and L is a maximum after taking into account the above condition.

The spectral terms, which relate to the different states, are thus denot ed by an approxi mation n2S + IT , where T is one of the letter s S, P, 0 or F for which L takes on respectively to signify one of the values 0, 1, 2, 3 . . .. If we take Cr+++ as an example, it has the configuration 3d 3 , and we find:

446

Optoelectronics of molecules and polymers

!

Smax = 3 x = ~ As I = 2 (sub-layer d), we have mr = -2, -I , 0, I , 2 and (L mt)max = (Mdmax = 2 + I + 0 = 3, wherein LMax = 3. By consequence, the fundamental state of Cr+++ is 34F (denoted simply as 4F).

-

2 Perturbation involving the coupling energy between different magnetic moments exactly tied to kinetic moments Two different cases can be distinguished: a when the spin orbit interaction energy is weak (typically for atoms with masses below that of mercury) In this case we use Russel-Saunders coupling, which is generally considered to not be significant until we have coupled it up with, on one side, all the orbital kinetic moments and, on the other, the total spin kinetic moment s. It is only after this that ~he tw~ kinetic moment s interact through the intermedi ate of spin orbit coupling. L and S represent the resultant of this total couplin g. The pert~r~ation Hamiltoni an due to the spin-orbit coupling is in the form Hs.o. = )"(L, S)L.S in which )"(L, S) is positive in the first half of the layer. With 1 = L + S ....L - S, there is a rise on the degeneration of the energy levels which is given by the proper values of the perturbation Hamiltoni an Hs.o.:

~E =

S 2 )..(L'2 )1i [1(1 + I) - L(L + I) - S(S + I)].

The spectral terms are thus denoted n 2s+ ]TJ, in which T corresponds, as above indicated, to the letters S, P, D and F which as values of L are, respectively, 0, I , 2, 3 . ... The upper left indice gives the number of possible orientations of the resulting spin, and is called the multiplicity of the term (doublet, triplet and so on), however, one needs to be prudent in using this notation. If S < L, then 1 takes on 2S + I and the number of values for 1 is given by the upper left indice, however, if S > L, then 1 cannot be negative and can only take 2L + I values and therefore there are only 2L + I permitted levels. In the case of Russel-Saunders coupling (also termed L - S coupling), the schematic representation of increment s in degeneration of energy levels under the influence of increasingly small perturbations are shown in Figure 3. For example , from the level L'S' degener ated [2L' + 1][2S' + I] times (resulting from exact electron interaction s, also called electron correlations) the spin orbit coupling is partially raises the degeneration so that I ' takes on [2L' + I] or [2S' + I] values (denoted as the values 1; , 1~ , 1~ in Figure 2). b Heavy atoms-jj coupling When the spin and orbital kinetic moments of each electron are coupled in the first step (with the resultant kinetic moment ji for the ith electron again obtained from

A-8 Energy levels and permitted transitions (and selection rules) in isolated atoms

447

L'" S"' L" S II L' S' J~ L'S' J~

Y(n,l) L'S'

L' S' J~

configuration LS

Spherical potential

Electronic correlations (Hee)

Spin orbit interaction

Figure 2. Levels of degeneration in L - S coupling .

terms of the Clebsch-Gordan series) , the resultant kinetic moments of each electron are then coupled between each other (jj coupling) to yield a global moment J. For an atom with 2 electrons with kinetic moments jj and j2, J takes on the values J =

Ijl - j21 ,··· .i. + j2. 3 Selection rules There are two final cases to consider:

a Russel-Saunders coupling for light atoms with several electrons The emission or absorption of light, which requires a variation by one unit of the kinetic moment, can happen when (selection rules) : •

LlS = 0 and the optical wave does not affect the internal movement of the electrons;

• •

LlL = ± I, 0 with Lli = ± I (l is the kinetic moment of an electron which undergoes the transition, while L is the kinetic moment of the atom ; LlJ = ± I , 0, however, the transition J = ~ is forbidden.

° °

b jj-Coupling for heavy atoms with several electrons The Russel-Saunders coupling selection rules are not applicable to heavy atoms with several electrons as the Sand L quantum numbers no longer have any physical significance, as the effective jj coupling suppresses the coupling of orbital moments to give L, along with the coupling of spin kinetic moments which give S. In this eventuality, transitions between singlet and triplet states S = 0 ~ S = I, where LlS i= 0, become possible (jj coupling for heavy atoms) . However, the condition LlJ = ±1 , 0 with the forbidden transition J = 0 ~ 0, remains [page 372, Cag 71].

A-9

Etching polymers with ion beams: characteristics and results

I Level of pulverisation (Y) 1 Definition y = (Mean number of ato~s .ej ect~d from target) number of incident IOns for a given ~t

In effect Y = f (E, J, e, M I, M2), in which M2, E, J and e are, respectivel y, the mass of the target atom , the energy of the incident ions (MI), the incident ionic current density and the angle between the incident ion beam and the normal to the target surface. With : • •

the energy transmitted to B during a sequence of colli sions (TB); the energy of the surface bond (Us) reason ably estimated to be equal to the sublimation of the target, so around 5 to 10eV,

the condition for ejecting an atom from the target is that TB ~ Us. There are two possibl e sequen ces: For the threshold energy, either ESI for sequence I or E SII for sequence II, and with the energ y of displacement as Ed, and 2M]M2 Y= (M] +M2)2 '

450

Optoelectronics of molecules and polymers

y

- equence I efficient for e =0 (normal incidence )

- Sequence II efficient for 11 ~ M2.

ESt

/

/rl T)'p~

Es" o T.

I collisio n sequence

Typ« /I collision

~qu. na

it is possible to write that

and

2 ESII = - (2Us y

+ Ed)

2 The result Yphysicat = f'(E): 3 zones Y (pulverised atom s/incident ions )

(IIy -(I)

() E,

I: below threshold (T B ::; Us ); II: E ~ E s and threshold energy fixed by sequence type I or II (TB ::: Us ); and III: when E > Es, rapid increase in Y, then a saturation plateau (reached rapidly if mass M I and M2 are grea tly different).

A-9 Etching polymers with ion beams: characteristics and results

451

Analytical forms (Yphysical): Thomson :

N2/ 3

Pul verisation level ~lhy.~i...· '" (atom/ion)

1

Us 1 + M2 cos O

:

-'. I

MI

I

cos 8

1

Y ex: - - ----=--:c-

"

(N : target atom ic den sity) Sigmund: Y(E, Us) ex: SN(E)/Us

1

Y(O)

Y(9) ex: - cos" 9

Anglen 50'

90'

Result ifMI t , Y t : if9t ,Y t: well verified when 9 < 60 0 •

3 Level of chemical pulverisation -

-

When using RIBE or IBAE, we can write: Ytotal = Yphysical + Ychemical If the projectile (P) forms with the target (T) a compound (TaPb) which is volatile, then in general, we observe that Ychemical S {; and the sign S takes into account the reflection of particles and the re-pul verisation of impl anted atoms ; If P and T form a non -volatile compound TaPb, we obser ve a decre ase in the velocity of attack on the target, so that Ychemical ;:: - (a~b) Y physical; When 0.25 < E < 2.5 keY, we tend to observe Yehemieal >- .JE; It has been observed that Ychemical ex: co s 9, so if 9 increases then Ychemical decreases.

II The relationship between etching speed and degree of pulverisation 1 At normal incidence

4

Surfa ce S

If N is the atomic den sity of the target and the volume eroded is equal to S x e (see Figure to left) then the number of atom s pulverised is N x S x e. During time At , the number of ion s incident to the surface S is equal to the flow of ions (D) multiplied by S. With D = J ~t in which q is the charge on mono. q charged IOns, NSe

Nq e

Nq

Y = - - = - - = - V. srt:.t/q 1 t:.t 1 With V = ~t we have the etching speed. In addition, it is po ssible to wr ite V = ~ Yl ex: Yl .

452 Optoelectronics of molecules and polymers

2 At oblique incidence The number of atoms pulverised is equal to N x S x e x cos 8. During time ~t, the number of ions incident to the surface at S is equal to D multiplied by S x cos 8. The result is again Y = !j9. ~t. By defining V(8)

ej

= -~t =

e· cos 8 ~t

,

we have:

,

Surface S

.

Y(8) = Nq V(8) J cos and we can also write: V(8)

1 = -Y(8)Jcos8 ex: YJ. Nq

Finally, we can always apply V ex: YJ when dealing with: -

non-reactive dry etching Y = Yphysical ; RIBE or IBAE in which Y = Ytotal = Y physical

+ Ychemical.

III Speed of reactive etching (IBAE Ar" /02 or 0+ /02) Degree of resurfacing (or fraction of absorbed surface): 8 = :;. , in which N is the surface concentration of absorbed molecule s and N* is the surface concentration of absorption centres. r is the flux of 0 2 and J is the flux of incident ions, so the flux of 0 particles absorbed is 2r1] (l 8) in which 1] is the probability of absorption for a molecule which hits an absorption centre . The degree of 0 atoms pulveri sed consists of 2 terms: •

•

0 atoms which have not reacted and is written a 8J, in which a is the level of pulverisation of 0 atoms which have not reacted and 8 the degree of absorbed surface; 0 atoms which have reacted and is written ~8J , in which ~ is the degree of reacted 0 atoms pulverised.

A-9 Etching polymers with ion beams: characteristics and results

453

At equilibrium, the number of absorbed 0 atoms equal s the number of pulverised o atoms and: 2fT] (1 - 8) - (ex . 8 . J + ~ . 8 . J) = 0, so

8=

ex

+~

(J) J« if

f , 8 = 1.

1+-- 2T] r

The etching speed is the sum of 2 components. One is physical and the other is chemical, each taking on the form Y x 1 in which J is the flux of incident ion s. For a unit surface (S = 1) of a given substrate, 8S = 8 take s on the chemical pul verisation while (l - 8)S = (l - 8) take s on the phy sical pulverisation. Thus:

IV ex Y chemical eJ + Y physical (l -

8)1

I

If the current of ions under study is 0+, we have to take into account the 0 + ions which attach themselves to the outer atom layers to yield reaction products. If y is the probability of an 0+ ion being fixed (implantation doe s not bring into playa 8-type coefficient for the absorbed surface), the contribution from the 0 + ion beam to the resurfacing of the target with oxygen is 2yJ.

The equilibrium equation becomes 2yJ + 2fT] ( I - 8) - (ex . 8 . 1 + ~ . 8 . J) With Q = rand

T]

= O.

<1> = ex + ~ , 2T]

we have J 1 +Q-

r 8=--7 J 1+<1>-

r

and retain:

IV ex Y chemical 8J + Y physical (l

- 8)1 I

If the reaction products are of the form XO n (in which, for polymers X is principally carbon), the probability of desorption being ~ (through pulverisation), and the number of 0 atoms which carry awa y X atom s is l/n, then the level of pulverisation of X atom s is: Y chemica l = ~ / n. From V we can deduce Ychemical, if Y physical is determined using an additional method, such as, for example, TRIM or empirically using a beam of Ar " ions .

454

Optoelectronics of molecules and polymers

IV Preliminary modelling of Yphysical for PI 2566 (a polyimide commecialised by Dupont and dedicated to microelectronics) For its use in microelectronics, PI2566 has a chemical formula which tends to favour etching, and can be used as an example to indicate typical etching parameters.

1 Levels of carbon pulverisation using 0+ ions Here the calculation is made on the basis of the surface bond strength (Us) being 7.4eV and with 2000 shots : • •

the chemical effects are not taken into account, so here are only cascading collisions or rather, physical effects; the level of pulverisation is at a maximum when E ~ 11 keV and e = 80° (Ymax ~ 1.36 atoms Cyion 0+). 1,4 ..---

-

-

-::::::;;;;;;r

-

-

-

-

--,

1.2 1

0.8

~:~ ~~~~~~~ 0.2 i

o

80 60

45 30

Incidence angle

e 15

'U

(0)

Ion energy (keV)

2 Comparing simulations of Yphysical (9) = f (9) and the Thompson and Sigmund models The various models show good agreement with TRIM obtained results. 45 , --

40

35

6'

30

-

-

•

TR IM (E

-

- - - - - - -- - - --

= 15 keY)

-

-

-

-

-

----,

- 0 - Thompson

Sigmund

>:'25 a> 20 >:' 15

......

10

5

o ~!=f:=-=:f=!!'==t=!=:t=!~~~~=-_+______l o 10 20 30 40 50 60 70 80 90 Angle of incidence

e (0)

A-9 Etching polymers with ion beams: characteristics and results 455

V Results from etching of polyimides 1 Self-supporting polyimide: the UPILEX Thi s material pre sent s high losses in opti cal absorption most likel y due to added pla stifying agents.

a Etching conditions The accelerating base energy was configured with the following optimised, and rather critical, parameters: (oxygen ions) E = 30keV,D = 10 17 ions cmI .J = 2I1-Acm- 2.

b Result A reasonably good result was obta ined in that an etching with depth 411-m was realised.

,

~

. . -

~ x 4 . 0 k 0012

10kV

leul

:--.l

c Nature of the etching With 0 + ions, there is no orientation effect from the beam (Sigmund or Thomson type laws are unverified for Yph ysical). With Ar" ions, and the same ion beam parameters, etching to 100 nm only is possible . If D ::: 10 17 ion s cm- 2, for 0+, the depth of the etching decreases a,d the polymer changes its appearance (due to a repeated depositing phenomenon).

456

Optoelectronics of molecules and polymers

d Conclusion Reactive etching due to oxygen chemi cal effects is extremely sensitive to the flow rate and can even greatl y influenc e the structure of the polymer, determining whether or not etching can take place. This test material , developed for its electrical properties, is not particularly well suited to optical applications, due to its high absorbenc y. With fluorinated polyimide s (for example Dupont's PI 2566), which are well suited to etching, more satisfying results were obtained (see Chapter XII, Section V).

2 A study of the etching of PI 2566 a Results from etching under accelerating conditions The parameters used come from those optimised for Upilex.

Nature of the ions 0+ was used with E = 30 keV, J = 211A em - 2, and the flow (D) was varied. In contrast to Upilex, the depth of the etching was insensitive (more or less constant) and remained relatively poor at e ~ 90 nm. Observation At D > 2 X 10 17 ionscm- 2 , the polyimide was degraded and could not be measured . Bubbles appeared due to strong local heating.

1400

~

1200

al 1000

..c B

"'" "'" "'"o"

is

•

800

•

•

•

600 400 200

0 0

5E+ 16

1E+17

1,5E+17

2E+ 17

2,5E+ 17

3E+17

Flow (ion crrr")

b Why? With 0+ ions, and also with Ar+ or He+ ions, a study of the effect of energy shows that e ~ Rp (see Figure below). In addition, under the ion beam the polyimide is very fragile and there is a formation , from the start of the irradiation (at E > 10 keV) of a hard layer with e ~ Rp (due to C = C and C = 0 bond ruptures and in consequence reticulation and contraction of the polyimide leading to e = Cte) which: -

limits (blocks) physical pulverisation (e ~ constant while D increa ses); and decrea ses the reactive polymer component in the polymer, which becomes just a thick carbonised layer.

A-9 Etching polymers with ion beams: characteristics and results

457

2500 . , - - - - - - - - - - - - - - - - - - - - - - - ,

•

Etched depth

2000

.g

.se,

.g 'E

]

1500 1000

500

0-+----+----+----+----+----+---+----1 10

20

30

40

50

60

70

80

Energy (keY)

c Conclusion These results indicate the import ance of keeping close to the RIE parameters (with an increase in J and a decrease in E which is of the order of only a few keV). Results obtained using parameter s close to the RIE are detailed in Chapter XII, Section V.

A-10 An aide-memoire on dielectrics: a model for double potential well and its application in determining the depth of trap levels

I Definitions of various dielectric permittivities [And 66] and the classification of different dielectric phenomena [Fre 60] 1 Absolute permittivity Generally, absolute permittivity (E) of an isotropic , dielectric material is defined as a quotient of an electrical induction formed by an electric field, so E= ~ . If we consider a charge (Q) caused by a potential difference (V) between the arms of a flat capacitor with surfaces S placed at a known distance (d) apart, the preceding ratio can be formed as: Cfreal Q/S Cd E= = = E V/ d S ' in which C is the capacitance. By consequence, in the meter, kilogram and second (MKS) system, the absolute permitti vity can be expressed in F m- I . If the system under study is a vacuum , then the vacuum permittivity (EO) is EO= 3Jn 10- 9 F m- I under MKS conditions, while in a centimeter, gram and second (cgs) system, EO= 1.

2 Relative permittivity It is often practical to introduce the term relative permittivity (s.) , which is also simply called, in an abuse of terms, permittivity . It is defined by the relation ship E= ErEOin which Er is a numbe r without dimen sion and equal in the ues system to absolute permittivity (or the dielectric constant), which explains the often encountered confusion between these two parameters. When an alternating tension (V = Voe j wt ) is applied to the flat condenser, which has a capacity Co when with an empty dielectric , the current which flows between the two plates on inserting a perfect dielectric (does not exhibit a leakage current) with permittivity Er can be written as

Q

1* = d . With dt

Q = CV = ErCOVod w t we have 1* = jWErCoV.

460

Optoelectronics of molecules and polymers

3 Complex relative permittivity As usual , in reality, the dielectrics in use are not perfect. They exhibit various sorts of currents, which we shall not detail for the moment, but give rise to a variety of similar effects. (Suffice to say that the effects of 'free' or 'bonded' carriers can give rise to heating effects or dielectric 'losses'). Such a current flow is entirely due to the dielectric material and we can characterise it with an imaginary component of the relative permittivity in the expression E; = E; - jE~ . In order to simplify the notation, the indices 'r' are often left out, and the reader is left to understand whether the system under study is working in relative or absolute permittivities. The intensity of the current thus becomes:

1* = WE~CO V + jWE;COV

= IR + jIe

I

The second term corresponds to a dephasing of between the tension and the purely capacitor based current. E; Characterises therefore the capacitor (insulator) part of the dielectric. The first term, relating to the fact that V and the intensity are in phase, corresponds to the resistive quality of the dielectric and is thus characterised by E~. The power dissipated by the Joule effect is thus written :

I

PJ = "2VoIo

I " 2 = "2COErWVo;

and the quantity E~ is called the dielectric absorption as it intervenes in the expression for the conversion of electrical energy to heat in the medium, thus absorbed by the dielectric and lost to the electrical circuit. liRI WE"CoV E" . = The quantity tan 1) = - .- = Ilel WE/CoV E/ is the (dielectric) loss tangent and allows a definition of the loss angle (1), which translates the dephasing between the resulting current and the 'ideal capacitor' part (Ie) of the current.

IR-~ The quantity Q = ~ s is a quality factor of the capacitor and is as large as tan 1) is small.

4 Limited permittivities Es is defined as the limiting (occasionally relative) permittivity obtained at low frequencies (the indice 'r' is removed by convention). At zero frequency we have a static field and Eoo is the (relative) limiting permittivity at very high frequencies (infinite frequency) .

A-1O An aide-memoire on dielectrics

461

5 Dielectric conductivity On taking the expressions E

= V/ d and Co = €oS/ d into 1*,

With 1 = aE and 1= ff 1 dS = IS , we can go on to the complex relation : 0'*

=

1 E =

1* " ES = W€o€r

+ JW€o€r . ,·

The real component for conductivity, called here the dielectric conductivity, is thus ad

= W€O€~ .

6 Classification of divers dielectric phenomena [Fre60] As already noted, the introduction of an imaginary term (10") into the dielectric permittivity, to take into account the existence of a current in phase with the applied tension , makes no presumptions about the origin of this current, whether how it is conducted or what losses may occur. In fact, there are numerous possible origin s of this dielectric absorption and we can class them, schematically, with respect to the frequency or temperature domain in which they appear : I absorption due to free carriers, for example , thermally generated carriers in the permitted band which exhibit a very low density ion insulators due to the limiting large forbidden band in these materials , or ions which permit ionic conduction at low frequencies; II the Maxwell-Wagner effect, due to charge accumulation in the discontinuities of the dielectric, found most notably in powders ; III the dipolar Debye absorption which is caused by bonded tied carriers and electrical dipoles; IV anomalies in 10' and 10" due to orientations of foreign molecules such as water at the solid surface; V anomalies in 10' and 10" tied with changes in phase ; and VI absorption due to resonance. Finally we present the Freymann representation [Fre 60] in Figure I below. Here the parameters of frequency and temperature are coupled. For a classical representation of 10' and 10" with respect to frequency only, we can turn to a textbook on electromagnetism, for example [May 96] or [Per 96]. Phenomenon I is due to a leak current resulting only from free charges, the number of which can be increased introducing impurities (such as ZnS, CdS doped with copper in analogy with semiconductors) into the dielectric. This results in an insertion of trap levels into the large forbidden band of the dielectric. Under a sufficiently high field, their electrons can attain a sufficiently high enough energy to collide with the lattice and provoke ionisations of the atoms . The upshot of this is that electrons are

462 Optoelectronics of molecules and polymers e

\A K 100Hz

1 MHz

e'

\:

A A\ 5

1Q MHz

V

~

High

temperatures

111

~

}j.

Low

temperatures

Figure 1. Representation of E' and E" as a function of l iT.

advantageously ejected into the conduction band to produce what is commonly known as a intrinsic breakdown of the dielectric. Phenomenon II is due to an accumulation of charges, such as electrons, in discontinuous regions within the dielectric. We can treat this problem as though studying a condenser made of several dielectric layers . Phenomenon III, otherwise known as a Debye dipolar absorption (DDA) generally results from the presence of neighbouring + and - charges (dipoles) which are linked so as to indisso luble, and act as pairs of tied carriers. The phenomenon is due to dielectric relaxat ions associated with the orientation of these dipoles once submitted to an electric field. The orientation occurs after a certain delay, characterised by a relaxation time . The dipoles exhibit a dephasing due to the inertia of their orientation with respect to viscous brushing against the rest of the material. We shall show in the following Section 2 how the hopping of a single charge over a potential barrier can be analysed following this sort of relaxation. Phenomenon IV can relate to a surface conductivity of the insulator, caused by 'semiconductors' such as dust, soilings and so on deposited on the insulator. This can also include surface humidity-impure water is a conductor with an extremely elevated dielectric constant of around 80. It is for this reason that the surface of a dielectric must be carefully cleaned prior to being studied . A usual methodology is to use alcohol first, to remove the humidity, and then a non-polar solvent such as benzene or toluene to remove the polar molecules, which could otherwise disturb the characterisation . And then of course , to work, if possible, in a dry atmosphere. Phenomenon V relates to an evolution in €' with changes in the dielectric density, caused by changes in phase , and above all, when the dielectric is a polar material, a degree of blocking of the dipoles in a new phase, for example when going from

A-1O An aide-memoire on dielectrics

463

the liquid to solid state. This mechanism can be accompanied by dipole 'frictions' between the dipoles and their environments giving rise to a heating effect in the dielectric which can be comparable to that due to a current. Phenomenon VI can be explained by considering that the charges of a dipole are elastically tied to an equilibrium position (in the simplest cases they can be compared to harmonic oscillators). Following the suppression of a field which displaced these dipoles from their equilibrium positions , they return to the equilibrium with a series of oscillations which display an amplitude which decreases as quickly as the breaking forces are strong. This model directly permits the possibility of an absorption due to resonance, which has a maximum at a characteristic alternating field frequency.

II Relaxation of a charge occupying two position separated by a potential barrier: dielectric model for double potential wells and its application to trap levels 1 Aide-memoire (see also Chapter IX, Section VII-2) The existence of trap levels in solids was demonstrated during the study of optical processes in semiconductors of groups II-VI. As an example we can take the spread of electronic levels in an inorganic phosphor such as ZnS(Cu), shown in Figure 2 and resembling that in Figure IX-14. Phosphorescence is due to a radiative emission which occurs with a delay after the initial luminous excitation, which excites electrons towards the conduction band (CB). The delay is caused by the passage of the same electrons, excited into the CB, through intermediates levels, called traps, in the forbidden band (FB). The relaxation from these traps is not instantaneous, thus the delay in the luminous emission and the persistence of phosphorescence, and occurs with a probability given by the Mott, Randall and Wilkins law, which states p = Vo exp( -U /k'I') , where vo is a constant for a group of traps . We can write t = l/p to detail the average trapping lifetime [Cur 60]. In contrast, fluorescence , caused by an instantaneous emission, results from an electron which does not pass through these intermediate trap levels.

CB

trapping detrapping (by thermal agitation of the crystal)

Trap U 2 FB

/v:-;;;;;

~hosPhorescence Figure 2.

464

Optoelectronics of molecules and polymers

Physically speaking we can characterise phosphorescence with respect to a thermal energy (kT)-loaned by phonons-which is involved in the transit of electrons trapped at an energy level A (the transition IA) towards another position with energyB by overcoming a potentialbarrierof heightV (the transitionlIB). The barrier corresponds to a trapping level E, with respect to the lowest part of the conduction band. Together, the transitions lA, lIB, and eventually, IIIp, make up the mechanism of phosphorescence. In addition, different trapping levels can exist, so we can have V, V2 and so on. 2 Transportation in a dielectric with trapping levels, and the effect of an electric field on transitions between trap levels a In the absence of an electric field

Here we shall look at an electron with two possible equilibriumpositions (A and B) separated by a distance a and a barrier V (assumed to be considerably greater than kT). The electron has as much chance of being found in A as in B, and the overall situation can be represented by a model of two potential wells separated by V, as in Figure 2-a. The probabilityper unit time for the electron to hop fromA to B (andreciprocally from B to A) is given by Poexp(- V /k'I'), where Po represents the same probability when the barrier is zero (i.e. when V = 0). At equilibrium, with N electrons per unit volume, all electrons of this type are equally spread between the two sites and the mean polarisation, due to these electrons, is zero. b When an electric field cib is applied in the sense AB (dielectrical measurements)

The potential energies VA and VB of an electron (of charge -q) differ by the quantity VA - VB = qaE. Given the orientation of the field, we have VA < VB, and by consequence(with V = -qV) VA is above VB by a quantity qaE, if the origin of the potential is taken as being at B (as in Figure 3-b). The symmetry of the system is thus broken, implicating a probability of presence of the electron being higher at B potential ener y

potential energy

distance

Figure 3.

A-lOAn aide-memoire on dielectrics

465

than at A, if we take into account the Boltzmann function which is proportional to exp (-U/kT).

c Expression for probability of transfer between wells Calculation of transition p robabilities Initiall y we assume that the electrons oscillate (with their therm al energy) about their equ ilibria positions with a frequency wo/2n . In the absence of a barrier, the probability that an electron will transfer from A to B or from B to A, in one seco nd, is thus wo/2n = Po. With a barrier of height (U - qaE ) on the A side and U on the B side, and in the presence of an electric field, these transition probabilities, respectively, PAB and PBA, become : Cl

PAB = _w_o exp ( __ u_-_q_a_E) 2n kT

(U)

PBA = -WQ exp - 2n kT

and

.

(1/)

If we suppose that qaE « kT, which typically is true for dielectric measurements taken under weak fields around I v cm'). we can go on to write: Wo UjkT (1 qa qa PAB=-e+E -) =PBA (1 +E -) . 2n kT kT

(I)

~ Variation in the number ofparticles in the A and B states at an instant t If at the instant t there are NA electrons in A and NB electrons in B, then there are NAPAB electron s which undergo the transition A ~ Band NBPBA elect rons which undergo the transition B ~ A, so that:

(2/)

On taking the difference between the above two equations and given that N = NA + NB is a co nstant and equal to the total number of electrons, then d (NB - NA) dt = -(PBA

+ PAB )(NB -

NA)

+ (PAB -

PBA)N .

(2)

On using eqn (I), PAB + PBA ~ 2PBA ( I

+ -qaE) 2kT

~

2PBA

(3)

and

qaE PAB - PBA ~ -PBA . kT Moving eqn s (3) and (4) into eqn (2) yields: d (NB - NA) qaE - - -- = -2PBA(NB - N A) + 2PBA- N. dt 2kT

The integration of this differenti al equation, without a seco nd member yields: NB - NA = C exp( - 2PBAt).

(4)

466

Optoelectronics of molecules and polymers

And moving this into the differential equation, while varying the constant C, gives: qaE

C = 2kTN exp(2PBAt) + K Byusingthelimitingconditionswhicharethatatt = O,NA = NB = N/2,determines that K = -i~N, from which can be derived that: NqaE NB - N A = - - ( 1 - exp[-2PBAt]). 2 kT

(5)

3 Expression for the polarisation at an instant t following the displacement of electrons a General case for displacement 8j in a system of qi charges at a concentration nj Moving the charge qi by Sj is the same as superimposing a dipole Slti = qiSj on the initial state . Schematically : OJ ~.

+ qj

.

by displacingq; by OJ

initial state

OJ ~

+qi

final state

To arrive at the final state from the initial, it suffices to superimpose on the latter a dipole Sltj = qjSj, as shown in the scheme below : s, iij ii;

• qi

~

+

•

~

-qj

0

I

+qj

011; =qi 0;

I

• 0

~

+qj

Similarly we can consider that the displacement of the charge qj by Sj is the same as applying a dipole Slti = qiSi . The polarisation (dipole moment per unit volume) associated with the displacement of charges qi by Sj is the same as applying a dipole moment per unit volume equal to Pi = niqiSj, where ni is the number of charges qi per unit volume . If there are other (i) types of charges (qj) present, then the polarisation is thus P

= LPi = LniqiSi . i

b Polarisation due to the displacement of N electrons-per unit volume-shared over a pair of wells with initial potential depth of U The two wells are separated by a distance a and, at an initial instant t = 0, are equivalent so that NA = NB = N /2 . Under the influence of a field E, the electronic charges

A -lOAn aide-memoire on dielectrics

467

(-q) are displaced by a distance +a for those in A and a distance -a for those in b, so: P

= Lniqioi = NA(-q)(+a) +

NB(-q)(-a)

i

(6)

= Ngqa - NAqa = qa(NB - NA)

Finally, on moving eqn (5) for (NB - NA) into (6) we acquire: N q2 a2 E P = 2kT(I - exp[-2PBAt]).

(7)

4 Practical determination of potential well depths a Basic formula Following on from eqn (7), if t -+

00,

P -+ Ps (static polarisation) we have:

N q2 a2E Ps = - - - . 2 kT

(7')

The term exp(-2PBAt) from eqn (7) describes the transitory regime of electrons trapped in potential wells and delayed (dephased) with respect to the applied field. As this varies exponentially with time, the relaxation of the system can be characterised by a macroscopic relaxation function: Y (t) = exp( - tiL) . The model for two potential wells thus can be seen in terms of a relaxation, much like that found in Debye 's theory, which yields similar equations for dielectrics (Debye's equations). The equivalence of the two terms obtained from the two well model and in Debye's theory (where Y(t) = exp( -tiL) as has been detailed elsewhere [leo 0I]), yields: exp(-2PBAt) = exp ( -~) ,

.

with L

I

= --. 2PBA

With PBA given by eqn (I ') , we can deduce: 11: L = _e U/ kT = Loe U/ kT WO

(8)

The term LO = 11:1Wo represents the duration of a single oscillation, as in the example A -+ B -+ A. In addition, as U » kT, we can find L » LO. The representation of the Debye curves shows us that E" goes through a maximum at a frequency Ve which is such that WeL = 1. Taking into account eqn (8) for L, we thus have: I 1 = 211:veLoeU/kT , so that Ve = -2-e- U/kT 11: LO In practical terms, the above formula is used in the form: Comment

U Log V e = - - - Log 211:Lo . kT

(9)

468

Optoelectronics of molecules and polymers

b Empirical characterisation Theoretically and as shown in Figure 4, we should obtain a straight line if we trace the logarithm of the critical frequency against inverse temperature. The slope of which should allow us to calculate the height of the potential barrier and the ordinance at the origin should yield the value of 1:0. In practical terms, we can arrive at the values for 1:0 and, above all, U, as we have done below with the example of Alq3. The law is in the form Vc = A exp(- U IkT), so that Log V c = - U IkT Log A. For I IT = 0, we have Log vc = Log A. The ordinance at the origin gives Log A, from which A = 1/2Jt1:0. In addition, when T = To we have Log vc = 0, so that liTo = k /U Log A which permits a calculation for U using U = kTo Log A. U is in eV with k = 8.64 x 10-5 eV K- I molecule",

+

Log v, Log A

Iff

Figure 4.

4 3.5

ech C6 Eact=O.217eV

•

3

............2

.-..

•

2.5 2

•

1.5

•

0.5

o 0.0035

0.0037

0.0039

0.0041

0.0043

1rr Figure S.

0.0045

0.0047

0.0049

A-IO An aide-memoire on dielectrics

469

c Experimental representation using Alq3 [Jeo Olb] The result of empirical observations is shown in Figure 5, from which U = 0.217 eV. This value is in good agreement with quantum chemistry calculations and thermoluminescence measurements.

A-II The principal small molecules and polymers used in organic optoelectronics

I Chemical groups and electron transport As indicated in Chapter X, a particular chemical gro up can be associated with a certain type of electron transport. As examples, the amine and carbazole functions favour p-type (hole) transport, while oxadiazole and triazole function s tend to favour n-type (electron) transport . Table I details their structures and characteristics. In general , metal based complexes act as electron s transporters.

II Examples of polymers used for their electroluminescence 1 The principal emitting polymers

Table 1. Functiona l group s and their favoured electrical transport Amine

Functiona l grou p

~/

I Conduction type Example

p TPD

Carbazole

Triazole

Oxadiazole

UD .cs. .cs. 0

~

I

I

p

n

n PBD

472

Optoelectronics of molecules and polymers

Table 2. The principal polymer based emitters (note abbreviations are widely used) Material

Amax (nm)

fluorinated polyquinoline (PQ) poly(para-phenylene) (PPP) polyalkylfluorene poly(3-cyclohexylthiophene) (peHT) poly(para-phenylene vinylene) (PPV) poly[2-methoxy,5-(2' -ethyl-hexoxy) - 1,4-phenylene vinylene) (MEH-PPV) poly(3-octylthiophene) (P30T) poly(3-alkylthiophene) (P3AT)

450 465 470 555 565 605 690 690

~)M n

(d)~" S

Figure 1. (a) PPP monomer unit (the polymer emits in the blue part of the visible spectrum); (b) PPV monomer unit (exhibits yellow emission); (c) MEH-PPV monomer units (emission is yellow-orange); and (d) P30T monomer unit (emits red light).

2 'The' polymer for hole injection layers(HIL)

n ~ n

Figure 2. The repeat unit of poly(3,4-ethylenedioxythiophene), which is often mixed with poly(styrene sulfonate), abbreviated commonly as PEDOT-PSS, to give an injection layer(HIL) .

A-II The principal small molecules and polymers used in organic optoelectronics

3 Example of a polymer used in hole transport layers (HTL)

Figure 3. The repeat unit of poly(N-vinyl carbazole) (PVK).

4 Example of a polymer used in electron transport layer (ETL)

Figure 4. Repeat unit of CN-PPV.

III Small molecules 1 The principal green light emitting ligands

JX)

0)

R = H : 8-hydroxyquinoline (q) R = CH 3 : 2-methyl-8-hydroxyquinoiine (mq)

7-azaindole

~

R

~

N

~

#

OH

N H

lO-hydroxybenzo[h]quinoline (Bq)

Figure 5. The structure of the principal ligands.

N

473

474

Optoelectronicsof molecules and polymers

2 Principal electron transporting small molecules emitting green light

Figure 6. Structures of the molecules: beryllium bis(lO-hydroxybenzo(h)quinolinate) (Bebqj) ( Aphotoluminescence = 529 nm); magnesium bis(2-methyl-8-hydroxyquinolinate) (Mgmqj) ( Aphotolumin escence = 487 nm); and aluminium tris(8-hydroxyquinolinate) (Alq3) ( Aphotoluminescence = 504 nm).

3 Example electron transporting small molecules emitting blue light

Zn (bis salen-t), Q-o-zn_o-y CH==N-1CH,~N==CH

Q-o--~-~ CH=N f cH , t N=HC

Zn (his salen-n),

Figure 7. Structures of Zn (bis salen-njj and the particular case of zinc bis(2-hydroxylatebenzylideneamino) butane (Zn (bis salen-4h,) which displays Aphotoluminescence = 460 nm.

4 Example small molecules which emit red light Small molecules which emit red light are principally based on europium octahedral complexes in which ligands (L) derived from ~-dicetone or aminopyrazine saturate the co-ordination sphere of the lanthanide (M) along with phenanthroline,

A-II The principal small molecules and polymers used in organic optoelectronics

475

or two electron donating molecules such as triphenylphosphine oxide or dibenzyl sulfoxyde.

5 Examples of small molecules whichserve principally as hole injection layers (HIL)

Figure8. Structure of copper phtalocyan ine (CuPc).

6 Examples of small molecules servingprincipally in hole transport layers (HTL)

Figure 9. Structures of the molecules : N, N'-diphenyl-N, N'-(3-methylphenyl)-I , l '-biphenyl4,4'-diaminc (TPD); 4,4-bis[N -(l-naphthyl)-N-phenylamino]biphenyl (NPB) ; and 4,4' ,4/1tris[N -( l-naphthyl)-N-phenylamino]triphenylamine (TNATA).

476

Optoelectronics of molecules and polymers

7 Example of a small molecule servingprincipally to confine holes in 'hole blocking layers' (HBL)

Figure 10. Structure of bathocupuroine (BCP).

A-12 Mechanical generation of the second harmonic and the Pockels effect

I Mechanical generation of the second harmonic (in I-dimension) 1 Preliminary remark: the effect of an intense optical field (E W ) Non-linear effects occur when the force (with E = E W ) returnin g displaced electrons towards their equilibrium position is no longer of the form fr = -kx = -mw2x. It is the Coulombic force developed by nuclei, which is exerted between nuclei and electrons, that gives this returning force . Typicall y, these internal electric fields are of the order of 108 to 109 V em -1 , while the optical field developed from several MW em - 2 is only of the order of 104 V em - 1. When the optical field is no longer negligible with respect to the internal field, for example on the use of a laser, the returning force must be rewritten in the form fr = -kx + dx 2 = -mw2x + m&2x, and is thus privileged by an inharmonic potential (W, although commonly denoted by V): ~

~

F = -grad W = -grad V

===}

I 2 I 3 V = -kx - -dx .

2

3

2 Placing the problem into equations The inharmonic potential , which takes on the form V (x) = ~ kx 2 - tdx3, along with the corresponding returning force fr = -kx + dx 2 are schematised in Figure I. The response of the system, in terms of displacements, due to an optical excitation (electrical field E W = Eo cos cot) is governed by the fundamental dynamic law for which a displacement x(t) is formulated with respect to the various harmonic components.

478

Optoelectronics of molecules and polymers 1

1 2

(a)

1

Vex) = - Ja2 - - dx 2 3

Vex) =-Ja2

3

(b)

l[~

i, (x) = -h+dx2 = -maJo x +m02;;2

i, (;;) =-h

Figure 1. Representation s of: (a) harmonic potentials; and (b) inharmonic potentials, each shown with their corresponding forces.

With the system being non-linear, we have to be prudent in using our habitual imaginary notation (E W = Re (Eoe j wt )) , and have to work alway s with real functions. On denoting the conjugated complex by cc, we have

EW

= E cos cot = ~ (e j wt + cc) .

When we study the movement of electrons in a material using classical forms (in the linear regime with a returning force fr = -kx), we ~an bri~~ in all the applied force s in the system, which include the Coulombic force fc = qE , in which q is the charge under study (q = -e for electrons), the friction force (ft ) , which is such that f t = -mfv x = -mfx,andthereturningforcefr = -kx (see, for example, Chapter 4 of [May 96]) . The same route can be used in non-linear regimes, although we must take on board the new expression for fr , as in fr = -kx + dx 2 = -mw2x +

mS2x. The fundamental equation for the dynamics with respect to x can thus be written as: mx = F = fr + ft + fe, so as to give, on simplifying withm:

L

(1)

A-12 Mechanical generation of the second harmonic and the Pockels effect V(r)

(a)

479

= ~/ci'- - ~dx 3 2

3

(b)

I

second harmonic _ _-t-l_ _-+_ X

1

-I--~---,-r---+-~--_

P

-

Dec omposition

/

tim

Movementdue to

fundamental mode

continuous mode

E(t) = E cos <:ot

Figure 2. Displacement of a charge placed in an inharmonic potential (a) and its decomposition(b).

Under a permanent regime, x is thu s sort after in a form which displays its development with respect to the harmonic components:

x(t) =

~ (xo +

x le iwt + x2ei2wt + .. . + cc) .

(2)

To simplify the calculation we can take Xo = a (even if the term exists: see Chapter XII, Sect ion III , studies therein) . The resolution of the problem is sadly through long, laborious calcul ations , which involve inserting eqn (2) for x(t) into eqn (I) . The result is the startlingly long equation (3) :

_

~2 (XIeiwt + cc) _ 2

200

2

(x 2e 2

iwt

+ cc) +

i~r (XIeiwt + cc) + ioor(X2eiwt + cc) 2

i2wt + 002° ( Xl e + cc ) + 002° ( X2 eiwt + cc) + 002° ( x2ei2wt + cc ) -

~

(xie2iwt +2xlx2'e- iwt +xlx7 +X2X2' +2x lx 2e3iwt

qE ( 1.wt ) = 2m e +cc

+x~e4iwt +cc) (3)

480

Optoelectronics of molecules and polymers

3 Solving the problem

a First order terms (linear approximation) Identifying the coefficient to the e iwt terms in eqn (3) gives X]

qE =-

I

m (w6 - (

2)

qE

+ iwf

I

~ ---------

2wm (wo - co) + i(fj2)

(4)

t if resonating at wo = co The linear polarisation for a displacement xr (t) of the charge q is thus p(l) (t)

= Nqxj (t) = Nq ~l

(e iwt

+ cc) .

By identifying with

we obtain (w)

Xl

Nq

I

= mEo (w6 -

(

2)

+ iwf

(5)

(here of course we find again the expression for the dielectric susceptibility of a linear medium; see eqns (4-36) in [May 96]) .

b Second order terms In the same way as for the linear system, the identification of the coefficients for e i2wt allows us to determine X2, which is such that

xf

(term dx 2 in the relation fr = -mw6x + We can thus see that it is the term in m82x2 = -kx + dx 2 ) which generates (forces) the movement in 2w. On moving eqn (4) for Xl into our latter equation, we arrive at:

The second order susceptibility appears within the second order term of the polarisation, which is written by analogy to the linear polarisation as

A-12 Mechanical generation of the second harmonic and the Pockels effect

481

From which we can thereforededuce X~2w) : (2w)

X2

Nq3 S2

I

= 2m2Eo [( w5 _ ( 2 ) + iWf]2 [( w5 - 4( 2 ) + 2iwf] .

(7)

xi

In a similar manner to X~w), 2w ) presents both real and imaginary parts but does, however, have a denominator which exhibits double resonance, as in co = wo and ~ w=2' Comment J Here the charge under considerationis that of an electron, i.e. q and thus q3 = _e 3. We can thus rewrite eqn (7): (2w)

X2

=-

Ne3 S2 2m2EO

I

[(W5 - ( 2 ) + iWf]2 [(w5 -

4( 2 )

+ 2iwf]'

= -e, (7/)

Comment 2 In addition, we can establish the following relationshipbetween linear and non-linear susceptibilities, respectively, X~w) and 2w ) :

xi

~2

(2w)

X2 (W)]2

.[ Xl

(2w)

Xl

2 EO

= ~ = 1;(2w) 2N2q3

The parameter 1; (2w) , called Miller's paramter, is in fact practically identical for all materials: 1;(2w) ~ 3 to 8 X 109 SI for InAs, GaSb, GaAs, Cd'Ie, ZnTe and ZnSe. Comment 3 In this Section we have looked at the problem in one dimension only. If instead of consideringan incident wavewith only one component, we imaginethat it presents three components, then the second order polarisation will have components Px , Py and Pz , which generallycan be obtained by using all possible quadratic components of Ex, Ey and Ez, as in:

Ex2 EY2 E2z , in which X~(2w) is thus a tensor. EzE y EzE x ExE y

II Excitation using two pulses and the Pockels effect 1 Excitation from two pulses

While studying the mechanical generation of the second harmonic, we have shown, in a classicalmanner, that an excitationdue to a photon (E W ) can generate movements

482

Optoelectronics of molecules and polymers

with frequencies of co (movement Xl(tl) and of 2w (movement X2(t». Respectively, photon emissions from charges which move through XI (at a frequency co) and through X2 (at a frequency 2w) do so by emitting dipolar rays at the frequencies wand 2w. We can thus generalise this emission mechanism for an excitation in a system of two photons with pulsations w' and « " , We should allow for the emission detailed the Figure 3: For a system excited with a pulsation co, we can use the classic eqn (7) for the second order (relative) susceptibility as follows: for an excitation pulse at Wi, given D(Wi) =

w6 - wf + iWir, eqn (7) becomes: (8)

and , similarly, for a two wave excitation pulsating at w' and «", the pulsation harmonic (WT = w' + co") is associated with a second order susceptibility in the form:

= Nq302

(w'+w")

'1...2

I

2m2co D(w/)D(wl!)D(w'

+ wI!)

(9)

2 The Pockels Effect A particular case which draws interest is when there is an incident optical wave (with pulsation w' = co) and a second wave at low frequency. At the limit zero, for a static field (wI! = 0), the preceding eqn (9) becomes:

X

(w+o)

2

Nq 3 02 I --- 2m2co D(O)D(w)D(w)

(10)

This second order susceptibility depends only on the optical pulsation co. Nevertheless, a reduction to the single, varying polarisation term is inexact, as we shall see . If we write the expression for the polarisation using the more usual development (see Chapter XII, Section III-2) , we have pet) = Po + XalE(t) + Xa2E2(t) + ... , in which here E(t) E(o) + E(w)ejwt + cc and where Xa is the absolute susceptibility, we can see that the dependence of the susceptibilities with two pulses

=

(J/:J

I~:~

------' to"

- -•• 2m'

:::---. zo"

~

( ' + q mH) In . W hiICh P and q are Integers " pm

Figure 3. Possibleemissions from a system excited by pulses at 00' and wI!

A-12 Mechanical generationof the second harmonic and thePockels effect

483

(static and optical) involves some complications in the calculation. In fact, we end up writing that (with relative susceptibilities to the 1st and 2nd order of Xl and X2) pet)

r

= £0 {Xl [E(O) + E(w)e jwt + cc] + X2 [E (O) + E(w)ejwt + cc + ...}

(11)

We can quickly see that this description is completely ambiguous as, for example, the first term (XI) carriers both the static term (E(o) and the optical term (E(w)e jwt) . If we continue with the calculation, even given the previous remarks (and thus Xl = XI(0) w or XI = XI (co) == ) by notation) while limiting ourselves to the static term and the pulsation term co, we have:

xi

pet) = £O{[XI(O)E(o) + Xl(w)E(w)e

jwt]

+ [X2(0)E{0) + 2X2(0 + co)

x E(o)E(w)e jwt]} + ... = {£o [XI(0) + X2(O)E(o) ]E(oJl

+ {£O[X I(co) + 2X2(0 + w)E(o) ]

x E(w)e jwt} + .." which gives a polarisation of the form: pet) = Po + Po+w(t)·

(12)

We can see here that the polarisation varying in co (the term Po+w(t» depends on + co) and also on XI (co) . A polarisation with frequency co is thus in the presence ofE(o), so: jwt Po+w(t) = £0 [Xl (co) + 2X2(0 + w)E(O) ] E(w)e . (13)

W2 (0

The indice 0 + co for Po+w(t) indicates that the polarisation is independent of co in the presence of the static wave E(o) and the optical wave E(w). This polarisation is a linear function of E(w)e jwt although it is calculated in the presence of the static field E(o). Additionally, in the absence of the static field E(o), we have as polarisation due to the pulse co, Pw(t) = £O(£r(w) - I)E(w)e jwt. With (£r(w) - 1) = Xl (co) = n2 - I, in which n = n(E(o) = 0) = n., using the notations detailed in Chapter XII (Section IV-5) we can write Pw(t) = £0(n2 - I)E (w)e jwt = Xl(w)E(w)e jwt. If we now consider the expression which takes on this polarisation in the presence of a static field E(o), this supplementary polarisation changes the indice to the form n + ~n, and we find that pet) = £o [(n + ~n) 2 - 1][E(o) + E(w)e jwt]

= £o[(n + ~n) 2 - I]E (o) + £o [(n + ~n) 2 - I]E(w)e jwt = Po + Po+w(t)

so that:

Po+w(t) = £o[(n + ~n)2 - l]E (w)e jwt .

By identification of Po+w(t) in eqns (13) and (14), we obtain:

(14)

484

Optoelectronics of moleculesand polymers

[en + fj,n)2 - I] = [Xl (00) + 2X2(0 + w)E(o)], so that with XI (00) = n 2 - I and [en + fj,n)2 - I] ~ n 2 + 2nfj,n - 1. This gives us 2nfj,n = 2X2(0 + w)E(o). And this we arrive at the definitive equation A

zxn =

X2(0+W)E (0). n

(15)

The variations in the indice appear to be proportional to the applied field E(o). However, this qualitatively exact result for the dependence of fj,n on E(o) is very imprecise with respect to the function in n due to the presence of the term X2(00 + 0), itself coming from the rather ambiguous eqn (11). A more phenomenological and alternative method, which takes account of the dependence of the optical indice on the static polarisation field-in organic media-is proposed in Chapter XII, Section 5. We can though go on to exploit eqn (11) by remarking that from eqn (5):

N

XI(W)

I

q =-mEo D(co) 2

and that D(O) = 005 = Cte, and from which, after eqn (11), (w+O)

X2

2 2 N q 2~2 o m EO 2 = -2 2 -N 4 XI(w) 2 m EO q 00 0

' - v - " '-v-'

I

I

D(w)D w

D(O)

(which resembles Miller's relationship) &2100

= 2 2N 2 XT(w), 00 0 q If with XT(w) = (lOr - 1)2 = (n 2 - 1)2 ~ n4 (and assuming 1 to be negligible with w O respect to n 2), we have + ) ex n4 •

xi

If we move this law into eqn (15), we find:

n4

fj,n ex - E(o), n that is fj,n

I 3 = "lrn s,

where E(o) = Es, and r is a coefficient of proportionality.

Comment It is worth noting that X(2w) comes from the term of xT (resulting from the term for dx 2 from the inharmonic force f = -kx + dx 2) so as to relate to the pulsation 200 . w O + ) corresponds to a pulsation of co. We have shown in Chapter XII, Section IV, that we do not have to return explicitly to the inharmonic term, thus simplifying greatly our calculations.

xi

Bibliography

[And 58] P.w. Anderson, 'Absence of diffusion in certain random lattices', Phys. Rev. B109, (1958) , 1492. [And 66] J. Ander son, Les dielectriques, Dunod, 1966. [And 92] J.1. Andre , 'Physique elementaire du transport dans les milieux organiques ; applications aux polyrnere s conducteurs' ; Vol. 9 (Proprietes electriques des polymeres et applications) from the series 'Initiation a la chimie et la physicochimie macromoleculaires' published by GFP (Groupe Francai s d'Etudes et d' Applications des Polymeres), 67083 Strasbourg . [Ant 98] R. Antony, 'Re alisation et caracterisations optoelectroniques de diodes electroluminescentes a base de polymeres electroactifs et de materiaux moleculaires deposes avec I'assistance d'un faisceau d'ions' , PhD, Universite de Limoges, June 1998. [Ant 00] R. Antony, A. Moliton, B. Ratier, ' Effect of various ions on organic light-emitting diodes obtained by ion-beam-assisted deposition' , Appl. Phys. B71, (2000) , 33-41 . [Am 91] P. Arnaud, Cours de chimie physique , Dunod , 1991. [ash 76] N.W. Ashcroft and N.D. Mermin , 'Solid state physics ' , HRW International editions, Saunders College publishing , 1976. [Ath 96] L. Athouel, G. Froyer, M.T. Riou, M. Schott, 'Structural studies of parasexiphenyl thin films: importance of the deposition parameters', Thin Solid Films, 274, 1996,35. [Atk 00] P.w. Atkin s, ' Physical Chemistry ', De Boeck University, 2000 . [baI99] M.A . Baldo, D.E O'Brien, M.E. Thomson, S.R. Forrest, 'Excitonic singlet-triplet ratio in a semiconducting organic thin film' , Phys . Rev. B, 66(20) , 1999, 14422. [Bal 00] M.A. Baldo , M.E. Thom son, S.R. Forrest, 'High-efficiency fluorescent organic lightemitting devices using a phosphorescent sensitizer' , Nature , 403, (2000), 750. [Bao 99] Z. Bao, LA, Roger s, A. Dodabalapur et al., ' Polymer light emitting diodes : new material s and devices ' , Optical Material s, 12 (1999) 177. [Bar OOa] P. Le Barny, C.M . Bouche, G. Veriot, ' Materiaux organiques electroluminescents', Techniques de l'ingenieur, EI -830 (2000), p. 1-24. [Bar OOb] P. Le Barny et al., 'Application of organic electroluminescent materials in visualisation', C.R. Acad . Sci. Paris, t.1, Series IV, (2000), 493 . [Bas 93] H. Bassler, 'Charge transport in disordered organic photoconductors, a Monte Carlo simulation study', Phys. Stat. Sol. B175, (1993) , 15-56. [Bel 99] J. Bell 'Organic groups excite rare-earth emiss ions' . Opto & Laser Europe , 60, (1999) ,21.

486

Bibliography

[Ber 68] DJ. Berets, D.S. Smith, Trans. Faraday Soc., 64, (1968), 823. One can also consult: Liang-Tse, M. Jozefowicz, R. Buvet, 'Conductivite electronique des composes macromoleculaires conjugues' , in 'Chimie Macromoleculaire', vol I, p. 570, Ed. G. Champetier, Hermann, Paris, 1970. [Ber Ola] R. Van den Berg, OLE, 'Flexible display is suitable for mass production ' , 85, June 2001,p.7. [Ber 01b] R. Van den Berg, 'The future looks bright for flexible organic displays ', OLE mars 2001,82, (2001), 29-31. [Bha 98] J, Bharathan, Y. Yang, "Polymer electroluminescent devices processed by inkjet printing : polymer light-emitting logo", App!. Phys . Lett., 72(21) , 1998, 2660-2662. [Bla 67] A. Blanc - Lapierre , Mecanique statistique, Masson , 1967. [Blo 97a] P.W.M. Blom, MJ. de Jong , C.T. Liedenbaum, r.r. Vleggar, ' Device characteri stics of polymer light-emitting diode s' , Synth . Metals , 85 (1997) 1287. [Blo 97b] P.W.M. Blom, MJ. de Jong, M.G. van Munster, 'Electric field and temperature dependence of the hole mobility in PPV ', Phys Rev B 55(2), (1997), R656 . [Blo 98] P.W.M. Blom , MJ. de Jong, 'Electrical characterization of polymer light-emitting diodes ' , lEE J. in Quantum Electronics, 4(1) , (1998), 1077. [Bos 95] Ch. Bosshard, K. Sutter, Ph. Prete, r. Hullinger, M.Florsheimer, P. Kaatz, P. Gunter, 'Organic Nonlinear Optical Materials ' , Gordon and Breach , 1995. [Bra 96] D.D.C. Bradley, 'Electroluminescent polymers : materials , physics and device engineering' , Current opinion in Solid State & Materials science, 1, 1996, 789. [Bra Ola] CJ. Brabec, N.S. Sariciftci , lC. Hummelen, 'Plastic solar cells' , Adv. Funct. Mater., 11,1, (2001) , 15. [Bra Olb] CJ. Brabec, N.S. Sariciftci , ' Recent developments in conjugated polymer based plastic solar cells ' , in Electroactive Materials , Ed. lO. Besenhard, W. Sitte, F. Stelzer, H. Gamsjager, Springer, Wien, 2001. [Bra 04] A. Boulben 'Les cellules solaires organiques atteignent 5% de rendement' , L'Usine nouvelle, N°2901 : 37 (22 janvier 2004). [Bre 99] K'V. Brennan , The physics of semiconductors with applications to optoelectronic devices, Cambridge University Press, (1999) . [Bre 82a] lL. Bredas, RR. Chance , R. Silbey, G. Nicolas and P. Durand, J. Chern. Phys., 77, (1982), 371. [Bre 82b] J.L. Bredas, RR. Chance, R. Silbey, 'Comparative theoretical study of the doping of conjugated polymers : polarons in polyacetylene and poly( para-phenylene)', Phys. Rev.B, 26(10), (1982) , 5843 . [Bro 01] DJ. Broer, J.A. Van Haaren , c.w. Bastiaan sen, 'Progress in information displays based on functional polymers' , European Polymer federation , special issue, June 2001, p.34--43. [brou 01] B. Brousse, 'Caracterisations de films de polyimide fluore obtenus par VDP en vue de la realisation de guides optiques', Rapport de stage DEA . Universite de Limoges, 200 I. [Bul 01] V. Bulovic, M.A. Baldo, S.R. Forrest , 'Excitons and energy transfer in doped lumine scent molecular organic materials' , Ch 11 in Organic Electronic Materials, Ed. R. Farchioni and G. Grosso, Springer series in Materials Science, Springer, Berlin, 2001 [Bur 90] J.H. Burroughes, D.D .C. Bradley, A.R . Brown, R.N. Marks , K.D. Mackay, R.H. Friend , PL Bum, A.B. Holmes , 'Light-emitting diodes based on conjugated polymers ', Nature , 347, (1990), 539-541. [Bur 91] Burroughes J.H. & RH. Friend, 'The Semiconductor Device Physics', in Conjugated Polymers, Ed. J.L. Bredas & R Silbey, Kluver, 1991, p. 555-622. [Bur 94] P.E. Burrows & S.R Forrest , 'Electroluminescence from trap-limited current transport in vacuum deposited light emitting devices ' , App!. Phys. Lett., 64(17), 1994,2285-2287.

Bibliography

487

[Bur 96] P.E. Burrows et al., 'Relationship between electroluminescence and current transport in organic heterojunction light-emitting devices', J. Appl . Phys ., 79, 1996, p. 7991-8006. [Bur 97] P.E. Burrows, S.R. Forrest , M.E. Thomson, "Pro spects and application for organic light-emitting devices " , Current Opinion in solid state & materi als science, vol 2, 1997, p.236-243. [Bus 94] M.N. Bussac , L. Zuppiroli, 'Stability of transverse bipolarons in conducting polymers' , Phys. Rev. B, 49(9), (1994), 5876 . [Cag 71] B. Cagnac, LC . Pebay-Peyroula, Physique atomique, Dunod, 1971. [Cam 98] AJ. Campbell, D.D.C. Bradley, D.G. Lidzey, 'Space-charge limited conduction with traps in PPV light emitting diode s' , lApp!. Phys ., 82(12) , (1997) , 6326. [Cam 98b] AJ . Campbell, M.S. Weaver, D.G. Lidzey, D.D.C. Bradley, E. Werner, W Bruning, M. Schwoerer, 'Conduction and trapping in electroluminescent polymer', Proceeding conf. SPIE, San Diego , vol 3476 , 1998, p. 98-1 10. [Cam 99] I.H. Campbell and D.L. Smith , 'Shottky energy barrier and charge injection in metal /Al /rnetal structure', AppI. Phys. Lett., 74(4) , 1999,561. [Car 84) E. Cartier, ' Des ecrans plats a cristaux liquide s pour les terminaux ' , Electronique industrielle, 78, (1984) , 99. [Cha 98] SC. Chang et al., 'Dual-color polymer light emitting pixels proces sed by hybrid inkjet printing' , Appl, Phys . Lett., 73( 18), 1998, 2561 . [Com 85] P.G. Le Comber, WE. Spear, 'Doped Amorphous Semiconductors' , in Amorphous Semiconductors, Ed. M.H. Brodsky, Springer, Berlin, 1985, p. 251. [Coh 73] C. Cohen Tannoudji , B. Diu, F. Laloe , ' Mecanique quantique', Herman, 1973. [Col 98] B. Colombeau, Contract report , France Telecom , 1998. [Con 97] E.M . Conwell, 'Transport in conducting polymers' , in Handbook of Organic Conductive Molecules and Polymers, Ed. Hari Singh Nalwa, 1 Wiley, Vol 4, Ch I, 1997. [Cor 011 C. Comic, B. Lucas, A. Moliton, B. Colombeau, R. Mercier, ' Elaboration and characterization of 6FDA /MPDA poly imide-based optical waveguide ' , Synth . Metals , paper R/P 44 (2001) . [Cox 87] P.A. Cox , 'The electronic structure and chemistry of solids ' , Oxford University Press , Oxford, 1987. [Cro 61] G.A. Crosby, R.E. Whan, R.M. Alire , ' Intramolecular Energy Transfer in Rare Earth Chelates. Role of the triplet State ', J. Chern . Phys ., 34(3), (1961), 743. [Cuo 89] U. Cuomo, S.M. Rossnagel , H.R. Kaufman, 'H andbook of ion beam processing technology' , Noyes Public ation , (1989). [Cur 56] M. Curie and D. Curie, ' Questions actuelles en lumine scence cr istalline, Ed. revue d' optique theorique et instrumentalle', Paris, (1956) . [Cur 60] D. Curie, 'Luminescence cristalline ' , Dunod , (1960) , p.160. [Den 00] V. Dentan, M. Vergnolle, H. Facoetti, G. Veriot, ' Progress in molecular electroluminescent material s', C.R . Acad . Sci. Paris, t.l , Series IV, (2000), 425. [Dep 83] S. Depp and W Howward, ' Les ecrans plats,' Science, 187, (1993) , 78. [Des 91] F. Desvigne, Rayonnements optiques. Radiometrie - Photometric. Masson. Paris, 1991. [Ebel 92] KJ. Ebeling, ' Integrated optoelectronics', Spring er, 1992. [Ell 90] S.R. Elliott , 'Physics of amorphous material s', Second edition , Longman, 1990. [Ell 98] S.R. Elliott, 'The physics and chemi stry of solids ', J. Wiley, 1998. [Emi 86] D. Emin , 'Basic issue of electronic transport in insulating polymers' , Handbook of Condu cting Polymers, T.A. Skothe im Ed., M. Dekker, Vo!.2, Ch 26, 1986. [Fav 01] lL. Fave and M. Schott, personal communication. [Fie 00] D. Fichou , 'Structural order in conjugated oligothiophenes and its implications on opto-electronic devices', J. Mater. Chem., 10, (2000) , 571 .

488

Bibliography

[Fis 88] G. Fishman, 'Energie et fonction d'onde des semiconducteurs' , Monographie de physique, Les editions de Physique, 91944-Les Ulis, (1988). [For 65] Th. Forster, 'Delocalized Excitation and Excitation Transfer', in 'Modem Quantum Chemistry ' , Ed. O. Sinanoglu, Part. III: Action of light and organic crystals, Academic Press, 1965. [For 98] E.W. Forsythe, D.C. Morton, c.w. Tang, and Y. Gao, 'Trap states in Alq3 using thermally stimulated luminescence ', Proceeding conf. SPIE, San Diego, vol 3476, 1998, p.123-130. [Fre 60] R Freymann and M. Soutif, 'La spectroscopie hertzienne' , Dunod, 1960. [Fri 70] H. Fritzsche . 'Les semiconducteurs amorphes' , La recherche, novembre 1970. [Fri 85] S. Fridrikhov and S. Movnine, 'Bases physiques de la technique electronique' , Ed. Mir, Moscou (1985). [Fri 92] RH. Friend, D.D.C. Bradley, A. Holmes, 'Polymer LEDs', Physics world, Nov. 1992, p.42-44. [Fri 98] RH. Friend, N.C. Greenham, 'Electroluminescence in conjugated polymers', in Handbook of conducting polymers, Dekker, Ch 29, 1998. [Gar 99] E Gamier, J. Zyss, 'Molecular Photonics for Optical Telecommunications : Materials, Physics, and Device Technology', Proceedings MRS 1998, Symposia 88, Elsevier 1999. [Gau 96] E. Gautier, J.M. Nunzi, C. Sentein, A. Lorin, P . Raimond, 'Blue light-emitting diodes with doped polymers', Synth. Met., 81, (1996), 197-200. [Ger97] M. Gerl and1.P. Issi, 'Physique des materiaux', Presses polytechniques et universitaires romandes , 1997. [Gho 74] A.K. Ghosh, D.L. Morel, T. Feng, R.E Shaw, C.A. Rowe, 'Photovoltaic and rectification properties of Al/Mg -phthalocyanine /Ag Schottky barrier cells', 1. Appl. Phys., 45 (1974),230. [Gil 72] W.D. Gill, 'Drift mobilities in amporphous charge-transfert complexex of trinitroftuorene and poly(n-vinylcarbazole)', 1. Appl. Phys., 43(12), (1972), 5033. [Gil 99] WP. Gillin and RJ. Curry, 'Erbium (III) lris(8-hydroxyquinoline) (ErQ): a potential material for silicon compatible 1.5 urn emitters' , Appl. Phys. Lett. 74, (1999), 798. [Gom 93] P. Gomes da Costa, RG. Dandrea, E.M. Conwell, 'First-principles calculation of the three-dimensional band structure of PPV', Phys. Rev., 47(4), (1993), 1800. [Gra 91] B. O'Reagan and M. Graetzel, 'A low cost, high efficiency solar cell based on dye-sensitized colloi'dal Ti02 films', Nature, vol. 353, 24 Oct. 1991. [Gre 73] G.N. Greaves, 'Small polaron conduction in V20S-P20S glasses', J. Non-crystal. Solids, 11, (1973),427. [Gre 94] N.C. Greenham, R.H. Friend, D.D.C. Bradley, 'Angular dependence of the emission from a conjugated polymer light-emitting diode: implication for efficiency calculations' , Adv. Materials, 6, (1994), 491. [Gre 95] N.C. Greenham, RH. Friend, 'Semiconductor Device Physics of Conjugated Polymers' , ed. H. Ehrenheich, Academic Press, (1995). [Had 86] R.C. Haddon, L.E. Brus, K. Raghavachari, 'Electron ic structure and bonding in Icosahedral C60', Chem. Phys. Lett., 125(5,6), (1986), 459. [Had 00] G. Hadziioannou, P.E van Hutten, Semiconducting Polymers, ch 16: 'A model oligomer approach to semiconducting polymers ', Wiley VCH (2000). [Hal 95] J.J. Hall et al., Nature 376 (1995)498. [Ham 96] PJ. Hamer, K. Pichler, M.G. Harisson, RH. Friend, B. Ratier, A. Moliton, S.C. Moratti, A.B. Holmes, 'Optical studies of chemical doping achieved by ion implantation in PPV' , Phil. Mag B, 73(2), (1996), 367-382. [Ham 97] Y. Hamada, 'Chelate Metal Complexes as organic electroluminescent materials' , in 'Organic Electroluminescent materials and devices ' , Ed. S. Miyata, Gordon, Ch9, 1997.

Bibliography

489

[Hay 87] S. Hayachi, K. Kaneto , K. Yoshino, ' Quenching of photoluminescence in polythiophene films by electrochemical doping ' , Solid State Comm ., 61(4), (1987), 249-251 . [Hee 88] A.J. Heeger, S. Kivelson, J.R . Schrieffer, WP. Su, 'Solitons in conducting polymers', Reviews of Modem Physic s, 60, (1988) , 78 1. [HeI67] W Helfrich, ' Space-charge-limited and volume-controlled currents in organic solids', in 'Physics and chemistry of organic solid state' , Wiley 3, 1967, 1-65. [Hoi 59] T. Holstein, 'Studies of Polaron Motion ', Annals of Physic s, 8, (1959), 325. [Hor 00] G. Horowitz, ' Physics of organic Field-Effect Transistor' , in Semiconducting Polymers, Ed. Hadziioannou G., van Hutten P.E, Wiley VCH, Ch 14, (2000) . [hut 01] P.E van Hutten , G. Hadziioannou, 'The role of interfaces in photovoltaic devices ', in 'Molecular Materials and functional polymers', Ed. W J. Blau, P. Lianos, U. Schubert, Springer, 2001 , p. 129. [loa 98] A. loannidis, E. Forsythe, Y. Gao, M.W Wu, E.M. Conwell, 'Current-voltage characteristic of organic light emitting diodes' , Ap!. Phys . Lett. , 72, 1998, 3038. [lof 60] A.E loffe, A.R . Regel, 'Non crystalline, amorphous and liquid electronic semiconductors', Progress in Semiconductors, vol.d, Heywood and Co LTD, London, 1960. [JjoI97] P. Jolinat, 'Etude et reali sation de diodes electroluminescentes organiques; realisation de diodes blanches' . PhD, Universite P. Sabatier, Toulouse, December 1997. [Jon 83] A.K . Jonsher, 'Dielectric relaxation in solids', Chelsea Dielectric Press, London 1983. [Jon 99] C. Jone s, 'The evolution of liquid-crystal displays ', Opto & Laser Europe, 69, (1999),41. [Jeo 01) Y.S. Jeong, B. Ratier, A. Moliton , L. Guyard, ' UV-Visible and infrared characterization of poly(p-xylilene) films for waveguide applications and OLED encapsulation' , Synth. Metals, 9135 (2001) , 1-5 . [Jeo 01b) Y.S. Jeong, D. Troadec, A. Moliton, B. Ratier, R. Antony, ' Dielectric studie s of Alq; and transport mechanisms' , Synth . Metals, 9136 (2001) , 1-6. [Kai 89] A.B . Kaiser, 'Thermoelectric power and conductivity of heterogeneous conducting polymers' , Phys . Rev B., 2806 , B 40(5),1989. [Kao 81] K.C. Kao, W Hwang, ' Electrical transport in solids ' , Pergamon Press, 1981. [Kar 97] S. Karg, M. Meier, W. Riess, 'Light-emitting diodes based on poly(para-phenylenevinylene): I. Charge- carrier injection and transport', J. App!. Phys. 82, (1997) , 1951. [Kel 92] M.K. Kelly, P. Etchegoin, D. Fuchs , W Kratschrner, K. Fostiropoulos, 'Optical transitions of C60 films in the visible and ultraviolet from spectroscopic ellipsometry' , Phys . Rev. B, 46(8) , (1992) , 4963 . [Keo 98] N.B. McKeon , Phthalocyanines Material s, Cambridge Univer sity Press, 1998. [Kil 71] H. KiIlcsreister et H. Bassler, ' Exciton reaction at an anthracene/metal interface : charge transfer' , Chern. Phys. Lett., II, (1971) ,411. [Kim 98] J.S . Kim, M. Granstrom, R.H. Friend, N. Johanson, WR. Salaneck, R. Daik, W.J. Feast, E Cacialli et al., ' Indium tin oxide treatements for single and double PLEDs' , J. App!. Phys. 84(12), (1998), 6859 . [Kit 96] C. Kittel , 'Introduction to solid state physics ' , John Wiley and sons, Inc, 1996. [Kiv 88] S. Kivelson, AJ . Heeger, ' Intrinsic conductivity of conducting polymers' , Synth . Met. , 22, (1988) , 371. [Kra 98] A. Kraft, A.C. Grimsdale, A.B. Holmes , 'Electroluminescent conjugated polymers: Seing polymers in a new light', Angewandte Chern., 37 (1998),402--428. [Law 92] Del R. Lawson, D.L. Feldheim, C.A . Foss , P.K. Dorhout, C.M . Elliott , C.R. Martin , B. Parkinson, 'Near-IR absorption spectra for the Buckminsterfullerene anions : an experimental and theoretical study' , J. Electrochem. Soc ., 139(7), (1992) , L68.

490

Bibliography

[Led 95] 1. Ledoux, R. Pinsard-Levenson, 1. Zyss, ' Materiaux organiques pour les reseaux de communications optiques: de la molecule aux composants' . L'Ech o des recherches, n? 162, 4eme trimestre 1995, p. 35-48. [Lei 98] G. Leising, S. Tasch, W. Graupner, ' Fundamentals of electroluminescence in paraphenylene type conjugated polymers and oligo mers', Handbook of conducting polymers, Dekker, Ch 30, (1998). [Lem 95 ] U. Lemmer et al., 'Aggregate fluorescence in conjugated polymers' , Chern. Phys. Letters, 240 (1995) 373-378. [Lev 68 ] S.N. Levine, 'Electronique quantique' , Masson, 1968. [Luc 03] B. Lucas, A Moliton et al., 'Gravure ionique reactive et applications a la realisation de modulateurs de phase en polymeres' (in press). [Lun 00] M. Lundstrom, 'Fundamentals of carrier transport' , second edition, Cambridge University Press, 2000. [May 96] M. May, AM. Cazabat, Optique, Dunod, 1996. [Mar 94] R.N. Marks, J.J.M. Halls, D.D.C Bradley, R.H. Friend, A.B. Holmes, J. Phys. Condens. Matter. 6 (1994) 1379 [Mat 98] H. Mathieu , ' Physique des semiconducteurs et des composants electronique s', Masson, ( 1998). [Men 93] Reghu Menon, CO. Yoon, D. Moses, AJ . Heeger, 'Transport in polyaniline near the critical regime of the metal- insulator transition' , Phys Rev B, 48, 1993, 17685. [Men 98] Reghu Menon, C O. Yoon, D. Moses, AJ. Heeger, 'Metal - Insulator transition in doped conducting polymers', Chapitre 2 du Handbook of conducting Polymers, Second edition, Ed. T.A. Skotheim, R.L. Elsenbaumer, J.R.Reynolds, M. Dekker, New York, 1998. [Mer 95] J.P. Mercier, P. Godard, 'C himie organique' Presses polytechniques et universitaires romandes, 1995. [Mig 0 I] E. Mignard, R. C Hiorns, B Francois, 'Synthesis and characterisation of star copolymers consisting of fullerene and conjugated polyphenylene' , Macromolecules, 35(16) 2002,6 132- 6 14 l. [Mil 60] A. Miller, 'Impurity conduction at low concentrations', E. Abrahams, Phys. Rev., 120 (1960) 745. [Miy 97] S. Miyata and S. Nalwa, 'Organic electroluminescent materials and devices' , Gordon and Breach (1997) [MoI 8t] A. Moliton. 'Cours polycopie de physique atomique' . Universite de Limoges, (1981). [Mol 88] A. Moliton, 1.L. Duroux, G. Froyer, 'Aspects theoriques et experimentaux du dopage physique des polymeres electroactifs ' . Ann. Phys. Fr., 13, (1988), 26 1. [Mol 9 1] A. Moliton et at.: ' Proprietes electroniques et schernas de bandes dans les semiconducteurs amorphes' : I 'Concepts fondament aux' , A. Moliton, B. Ratier, Ann. Phys. Fr., 16, (1991), 261. II 'Etude des phenomenes de transport', A Moliton , B. Ratier , Ann. Phys. Fr., 16, (1991), 305. Ill ' Les mecanismes de transport par saut faisant appel aux modeles des paires et a la conduction par polarons', A Moliton, B. Lucas, Ann. Phys. Fr., 19, (1994), 299-352. [Mol 94] A. Moliton, B. Lucas, C Moreau, R.H. Friend, B. Francois, ' Ion implantation in conjugated polymers : mechanisms for generation of charge carriers', Phil. Mag. B, 69(6), (1994), 1155- 117 1. [Mol 98] A. Moliton, ' Ion implantation dopping of electroactive polymers and device fabrication' , Handbook of conducting polymers, Dekker, Ch 21, (1998). [Mol 00] A. Moliton, R. Antony, D. Troadec, B. Ratier, 'Ion beam assisted deposition of organic molecules: a physical way to realize OLED structures' , C R. Acad. Sci. Paris, t. I, Series IV, (2000), p. 437-446.

Bibliography

491

[Mom 95] B. Mombelli, 'Processus optiques dans les solides' , Masson , 1995. [Moo 93] E. Mooser, 'Introduction ala physique des solides', Presses poly techniques romandes, 1993. [Mor 78] D.L. Morel , A.K. Ghosh, T Feng , E.L. Stogryn, P.E. Durwin, R.F. Shaw, C. Fishman, App\. Phys . Lett., 32 (1978), 495--497. [Mor 97] C. Moreau, R. Antony, A. Moliton, B. Francois, 'Sensitive Thermoelectric power and conductivity measurements on implanted PPP films ', Adv. Mat. for Optics and Electronics, 7,281, (1997). [Mot 71] N.F. Mott, E.A. Davis, 'Electronic processes in non-crystalline materials' , Clarendon press, 197 I. [Mot 79] N.F. Mott , E.A. Davis, ' Electronic processes in non-crystalline materials', Second edition, Clarendon press, 1979. [Mot 931 N.F. Mott , ' Conduction in non-crystalline materials' , Oxford Science Publications, 1993. [Mou 99] C. Moussant, 'Gravure ionique reactive de type RIBE et IBAE . Application a la realisation de guides optiques polymeres' . PhD thesis ; l'Universite de Limoges, May 1999. [Mou 00] C. Moussant, B. Lucas, A. Moliton, 'Etching techniques using a collimated ion beam for the realization of waveguides', Synth. Metals, 115 (2000), 29. [Mill 89] K.H . MUlier, 'Film growth modification by concurrent ion bombardment: theory and simulation', Ch 13 in 'Handbook of Ion Beam Processing Technology' , Ed. 1.1. Cuomo, S.M . Rossnagel and H.R. Kaufman, Noyes Pub., (1989) . See also K.H . MUlier, ' Model for ion-assisted thin film densification' , J. Appl. Phys. , 59(8) , 1986,2803. [Nag 85] 'Electronic transfer in amorphous semiconductors', in Amorphous Semiconductors (M.H . Brodsky, Ed.), Springer Verlag, Berlin , 1985, Chap. 5. [Nag 97] K. Nagayama, T.Yahagi, H. Nakada, TTohma, T Watanabe, K. Yoshida , S. Miyaguhi, ' Micropatterning method for the cathode of the organic electroluminescent device' , Japan J.App\. Phys ., 36, (1997), L 15555-L 1557. [Ngu 01] T.P. Nguyen, P. Molinie, P. Destruel, ' Organic and polymer-based light-emitting diodes ' , Chapter 1 of vo\. 10, 'Handbook of Advanced Electronic and Photonic Materials and Devices' , Ed. H.S. Nalwa, Academic Press, 2001. [Ngu 94] Nguyen Trang Anh , 'Introduction a la chimie moleculaire', Ellipses, 1994. [nie 99] J.F. Nierengarten, J.F. Eckert, J.F. Nicoud, L. Ouali, V. Krasnikov, G. Hadziioannou, Chern Commun. (1999) 617 . [Nie 01] J.F. Nierengarten, G. Hadziioannou, N. Armaroli, 'Molecular photovoltaic devices', Materials Today, 4(2), 2001,16. [Nun 02] J.M . Nunzi, ' Organic photovoltaic materials and devices', C.R. Physique, 3 (2002) 523-542. [ole 00] OptoLaserEurope, vol 79, (nov 2000), 19-23. [ole 01] Market report, p. 47, ' Opto et Laser Europe' , vol 81, January j February 2001. [opt 93] ' Optoelectronique moleculaire' , ' Observatoire Francais des techniques avancees' , Arago 13, Masson, 1993. [Pan 71] J.1. Pankove , 'Optical processes in semiconductors', Dover Pub , 1971. [Pap 85] M. Papuchon, Y. Bourbin, S. Vatoux, ' Etat de I'art de l'optique integree' , ' Optique guidee monomode' , Masson, 1985, p. 621 references therein . [Par 94] I.D. Parker, 'Carrier tunneling and device characteristics in polymer light-emitting diodes ', J. App\. Phys ., 75(3) , (1994),1656. [Pei 55] R.E. Peierls, 'Quantum theory of solids ' , Clarendon Press , Oxford, 1955. [Per 96] J.P. Perez , R. Carles, R. Fleckinger, 'Electromagnetisme', Masson, 1996.

492

Bibliography

[Pet 91] R.A. Soref, 1. Schmidtchen, K. Petermann, 'Large single-mode rib waveguides in GeSiSi and Si-on Si02 ', IEEE J. Quantum Electronics, 27(8), (1991),1971 , and Electronics Lett., 27, (1991) , 1488. [Pet 96] G. Fischbeck, R. Moosburger, M. Topper, K. Petermann, 'Design concept for singlemode polymer waveguides ' , Electronics Lett ., 32(3), (1996), 212. [Peu 03] P. Peumans, A. Yakimov, S.R. Forrest, 'Small molecule weight organic thin-film photodetectors and solar cells', 1. App!. Phys ., 93(7), 2003, 3693. [Pfu 86] P. Pfluger, G. Weiser, Campbell,1. Scott, B. Street, 'Electronic structure and transport in the organic amorphous semiconductor polypyrole' , 'Handbook of conducting polymers ', Vo!. 2, Ed. T.A. Skotheim, Dekker, Ch 38, (1986). [Pic 95] K. Pichler, C.P. Jarrett, RH. Friend, B. Ratier, A. Moliton , 'Field-effect transistors based on poly(p-phenylene vinyIene) doped by ion implantation', J. App!. Phys., 77(7) , (1995) , 3523. [Pop 82] M. Pope, C.E. Swenberg , 'Electronic processe s in organic crystals', Clarendon Press, 1982. [Que 88] Y. Quere, 'Physique des materaiux ' , Ellipse s (Paris), 1988. [Ran 45] J.T. Randall and M.H.F. Wilkins, ' Phosphorescence and Iectrons trap distribution, 1- The study of trap distributions', Proc . Roy. Soc.,184A, (1945), 365. [Riv 89] 1.L. Rivail, 'Elements de chimie quantique' , InterEditionslEditions du CNRS (1989) . [Ron 98] J. Roncali , 'Advances in the molecular design of functional conjugated polymers' , Ch. 12 du Handbook of conducting polymers, Dekker, (1998) . [Rot 96] Rothberg L.J., Lovinger A.J., 'Status of and prospects for electroluminescence ' , 1. Mat. Res., 11(12),1996,3174. [Sal 91] B.E. Saleh, M.C. Teich, 'Fundamentals of photonics ', Wiley, 1991. [Sam 00] LD. Samuel, A. Beeby, 'Sidestepping the selection rules', Nature, 403, 2000, 710. [Sap 90] B. Sapoval , C. Hermann , 'Physique des semi-conducteurs', Ellipses, 1990. [Sch 94] M. Schott , M. Nechtschein, 'Introduction to conjugated and conducting polymers' , in Organic Conductors' , Ed. J.P. Farges , M. Dekker, p. 495, (1994). [Sch 95] A. Schmidt, M.L. Anderson, N.R Armstrong, J.App!. Phys ., 'Electronic states of vapor deposited electron and hole transport agents and lumine scent material s for light-emitting diode s' , 78(9), (1995), 5619. [Sch 00] Schott M., 'Introduction to the physics of organic electroluminescence', C.R Acad. Sci. Paris, t. 1, Series IV, (2000), 381-402. [Scho 00] J.H. Schon, Ch. Kloc, A. Dodabalapur, B. Batlogg, 'An organic solid state injection laser' , Science, 289, (2000) ,599. [Scho OOb] 1.H. Schon, Ch. Kloc, E. Bucher, B. Batlogg , 'Single crystaline pentacene solar cells', Synth. Met ., 115 (2000) 177. See also, 1.H. Schon, Ch. Kloc, E. Bucher, B. Batlogg, ' Efficient organic photovoltaic diodes based on doped pentacene', Nature 403, (2000) , 408 . [Seg 01] L Seguy, 'Realisation et etude de diodes electroluminescentes organiques abase de molecules discotiques', PhD, Universite Paul Sabatier, Toulouse, 2001. [Sen 97] C. Sente in, C. Fiorini , A. Lorin, 1.M. Nunzi, 'Molecular rectification in oriented polymer structure' , Synth. Metals, 91 (1997) 81-82. [She 98] 1. Shen, F. So., 1. Yang, J.H. Xu, Y.E. Choong , H.C. Lee, 'Carrier transport in light emitting diodes' , Proceeding conf, SPIE, San Diego , vol 3476, 1998, p. 196-201. [Shi 132] Y. Shirota , S. Nomura, H. Kageyam a, 'Charge transport in amorphous molecular materials ' , Proceeding conf. SPIE, San Diego, vol 3476, 1998, p. 132-141. [Shi 00] y. Shi et al., 'Low halfwave voltage polymeric electro-optic modulators achieved by controlling chromophore shape', Science, vo!. 288, 7 Avril 2000, p. 119-122.

Bibliography

493

[Si 97] J. Si and C.W Tang, 'Doped organic electroluminescent devices with improved stability' , Appl. Phys . Lett., 70(13) (1997) 1665 [Sic 00] L. Sicot, C. Fiorini, C Sentein , A. Lorin , P. Raimond, lM. Nunzi, 'Improvement of the photovoltaic properties of polythiophene-based cells', Solar Energy Materials & Solar Cells , 63, (2000) 49-60. [Sim 85] r. Simon, r.r. Andre , 'Molecular Semiconductors', Ed. lM .Lehn, Ch. W Rees, Springer, Berlin, 1985. [Sko 86] T.A. Skotheim, 'Handbook of conducting Polymers', M. Dekker, New York, 1986. [Sko 98] T.A. Skotheim, R.L. Elsenbaumer, lR. Reynolds, 'Handbook of conducting Polymers', Second edition, M. Dekker, New York, 1998. [Sly 96] S.A. Van Slyke , C.H. Chen , e.w Tang, 'Organic electroluminescent devices with improved stability', App\. Phys. Lett., 69(15), 1996, 2160 . [Smi 90] EA. Smidt, 'Use of ion beam assisted depos ition to modify the microstructure and properties of thin films' , International Materials Reviews, 35(2), 61 (1990). [Smi 61] R.A. Smith , 'Wave mechanics of crystalline solids', Chapman and Hall, 1961. [Su 79] WI'. Su, lR. Schrieffer, AJ. Heeger, 'Sol itons in polyacetylene' , Phys. Rev. Lett., 42, (1979), 1698. [Sut 93] A. Sutton, 'Electronic structure of materials', Clarendon Press, Oxford, 1993. [Tah 98] R. Tahar, T. Ban, Y. Ohya, Y. Takahashi, 'In doped oxide films: electrical properties', lApp\. Phys., 83(5), 1998,2631. [Tan 86] C.W Tang, 'Two-layer organic photovoltaic cell', App\. Phys . Lett., 48( 1986) 183 [Tan 87] C.W Tang et S.A. VanSlyke, 'Organic electroluminescent diode', Appl. Phys . Lett., 61, (1987),913. [Tes 96] N. Tessler, OJ. Denton , R.H. Friend, ' Lasing from conjugated polymer microcavities' , Nature, 382, 1996,695 . [Tes 98] N. Tessler, N.T. Harisson, D.S. Thomas, R.H. Friend, 'Current heating in polymer light-emitting diodes' , Appl. Phys. Lett., 73(6) , (1998) , 732-734. [Til 00] R. Tilley, 'Colour and the optical properties of materials' , Wiley, (2000) . [Tou 01] E. Toussaere, 'HDR', ENS Cachan, Univer site of Paris XI, 2001. [Tra 00] lP. Travers, ' Polymeres conducteurs: conductivite adiverses echelles' , Bulletin de la S.F.P., 124, (May 2000), 8. [Tro 92] N. Trouiller, J.L. Martins, 'Structural and electronic properties of C60', Phys Rev. B, 46(3), (1992) , 1754. [Tro 96] P. Trouillas, ' Le carbone 60: de l'origine de ses proprietes electroniques et optiques a son comportement sous faisceau d'ions', PhD n? 21-1996, Univers ite de Limoges. [Tro 01] D. Troadec , 'Contribution a l'elaboration et a la carcterisation de diodes electroluminescentes organiques' , PhD, l'Universite de Limoges, November 2001 . [Tro 03] D. Troadec, J.M. Nunzi, 'Caracterisation optoelectronique de cellules solaires photovoltaiques organiques C60 /CuPc (P-N) et C60 :CuPc (co-evaporees)' , Jounees Polymeres Conducteurs, A5-6 , Dourdans, Sept. 2003 . [Uet 86] M. Ueta, H. Kanzaki , K. Kobayashi , Y. Toyzawa , E. Hanamura, 'Excitonic Processes in solids' , Solid-State Sciences, vol, 60, Springer-Verlag , Berlin , 1986. [umop 99] A. Moliton, J.P. Moliton, lM. Dumas, Compte rendu d'activite du laboratoire UMOP (1999 /2000). [Wri 95] l D. Wright, 'Molecular crystals ' , Cambridge University Press , 1995. [Wu 96] C.c. Wu, J.C. Sturm, R.A. Register, M.E. Thomson, 'Integrated three-color organic light-emitting devices' , App\. Phys . Lett. 69(21), 1996,3117-3119. [Wu 97] c.c. Wu, e.r. Wu, J.e. Sturm , A. Kahn, ' Surface modification of ITO by plasma treatment', Appl. Phys. Lett., 70(11), 1997, 1348.

494

Bibliography

[Yar 89] A. Yariv, 'Quantum Electronics' , Third Edition, Wiley, 1989. [Yu 95] G. Yu, AJ. Heeger, 'Charge separation and photovoltaic conversion in polymer composites with internal donor /acceptor heterojunctions', 1 Appl. Phys . 78 (1995), 4510 . [Zab 84] A.G. Zabrodskii, K.N. Zinov'eva, ' Low temperature conductivity and metal-insulator transition in compensate n-Ge ', SOY. Phys . JETP, 59(2), Feb 1984,425. [Zup 91] L. Zuppiroli, 'Le fil moleculaire,' Rapport CEA-R-5543, Document service, CEN Sac1ay. [Zup 93] L. Zuppiroli, M.N. Bussac, S. Paschen, O. Chauvet, L. Forro, P. Bujard, K. Kai, W. Wernet, 'Hopping in conducting polymers ', Proc . 5th Int. Conf. on hopping and related phenomena, Ed. CJ. Addkins, A.R . Long , lA. Me Innes, Glasgow, 31 August - 3 Sept. 1993, p. 171. See also, L. Zuppiroli, M.N. Bussac, S. Paschen, O. Chauvet, L. Forro, 'Hopping in disordered polymers', Phys Rev. B, 50,1994,5196.

Index

absorption (spectral), 184,401 acceptor-donor, 195, 326 adjusting emissions, 285 ageing, 294 aggregates, 194 Alq3, 61, 69, 94, 259 amorphous silicon xi Anderson localisation, 47 anodes (OLEDs) 278 anthracene (narrow band) , 109 band curves, 261 forbidden, 6 HOMO, 65, 75 Hubbart's, 43 LUMO, 65, 75 scheme, 92, 96 structures, 33 barrier - at inte rfaces, 259 benzene (electron structure), 66 bilayer heterostructure, 324 OLED,280 photovoltaic cell , 322 bipolaron, 82, 86 Bloch (function), 395 Boltzmann (transport equation), 404 bond strong, 4, 9 weak , 3, 6 cathodes (OLEDs), 278 cathode-virtual, 139 cha in

of atoms , 12,26,389 distorted, 20 characteristic s I(V), 264, 271 charge transfer complex, 45 chromophore, 177 coefficient s - opto-electric, 341, 356 conduction-organic (n- or p-type), 276 conductivity, 99 conduction by hopping, 112 configuration curves, 87, 168 diagram of, 179 electron, 164, 443 conjugation (molecule/polymer), 62 contact at electrodes, 259 , 268 ohmic, 138 rectifying , 135 coupling between atoms, 39 between molecul ar orbit als , 40 electron-lattice, 77 Frank-Condon, 184,264 current bipolar, 157 equilibria, 277 saturation, with traps, 147 space cha rge limited, 139 volume controlled, 154 Davidov (displacement, breakdown), 192 defaults (effect s of), 90-93 depo sition, 201 assisted, 204 vapour phase, 202

496

Index

Dexter transfer, 197, 288 dimer,189 diodes inorganic, 236 molecular optic, 349 organic, 253, 257 white, 296 donor-acceptor, 346 dopants electrical, 126 optical , 286 doping, 87, 126, 431 Dushman-Richardson law, 136, 433 electric (transport properties), 99, 133 effective mass, 35, 396 electro-optic modulator, 358, 368 electron pumping, 292 emission electrical (laws), 135 optical internal/external, 221 encapsulation, 203 etching, 216, 449 ofITO,211 of pixels, 298 RIBE /IBAE, 366 speed, 451-452 excitation transfer, 286 exciton, 87, 185 charge transfer, 186 dissociation, 322 Frenkel, 188 Wannier, 186 fabrication (techniques), 201 fill factor, 318, 333 film, 204 film growth , 205 Floquet (theorem), 9 fluorescence, 182, 194-195 flux, 406 Forster (transfer) , 196, 289 Fowler-Nordheim (emission), 137,437 fullerenes, 61, 70, 397 gauss (trap distribution), 154 Graetzel cells , 320 guides diffusion , 232 fabrication, 213 losses, 233, 368

half-wave tension, 344 heterojunction OLEOs, 253 volume , 325 hopping nearest neighbour, 112 over variable distance, 113 polarons, 116 Hubbard (band), 43 hybridisation, 36, 381 hopping (see conduction by hopping) improving OLEOs, 274 infra-red OLEO s, 291 injection (level of) , 133 ink-jet, 305 ink-jet technology, 302 interaction dipolar, 196 intrachain, 66 interchain, 67 Ioffe-Regel, 48, 51 irradiation, 94 Jablonsky, 181-183 Kaiser's model , 129 Kerr effect, 341 Kubo-Greenwood, 403 laser (effect) , 292 layer confinement, 279, 476 emitting, 284,474 injection, 280,473,474 transport, 282, 473, 474 layering , 279 light amplifier, 173 limits conduction model with delocalised states band , 105-109 rigid band theory, 48 liquid crystal, 239 localisation (Anderson), 50 luminance, 220 measuring, 230 luminescence extinction, 248 Mach-Zehnder, 364 matrix active, 243 doped, 165 passive, 243

Index merit factor, 359 merit figure, 360 microcavities, 292 Miller and Abrahams, 52 mobility, 51, 101, 107-108,412---413 in Alq3 265-266 in polymers, 271 thermally activated, 421 molecule centrosymmetric, 349 non-centrosymmetric, 350 Mott (localised states), 54 Mott and Gurney (law), 140 MIM structure (metal-insulator-metal), 321 MIS structure (metal-insulatorsemiconductor), 252 multi-layers - see layering OLEDs, 253, 257 optical losses, 233, 368 orbitals - atomic and molecular, 373 page - flexible electronic, 304 Peirls (transition), 24, 64 permittivity - dielectric, 459 phases - interpenetrated, 324 photodiode (sensitivity), 226 photovoltaic, 195, 313 phosphors, 249 phosphorescence 182, 250 photoluminescence, 184 photometry (scaling) , 217 phtalocyanines, 316 pixels, 298 Pockels effect, 343 polaron, 77, 86 polaron - exciton, 88 polaron - transverse , 126 polyacetylene, 64 polyimide, 363 polymer degenerate -(CH)x-, 64 non-degenerate (PPP), 65 polymerisation in the vapour phase (VDP), 203 poly(para-phenylene) (PPP), 60, 65, 86, 94 poly(phenylene vinylene) (PPV), 60, 184 Poole-Frenkel (conduction), 159 pulverisation time, 449

quantum interferences, 104 rare earths , 164, 169-173 relaxation, 409 energy, 25 time, 417 rules circumnavigating (bending), 288 transition , 163, 176, 180,444,447 Schottky emission, 136, 434 screen electroluminescent, 246 liquid crystal (LCD), 239 micro-points, 245 organic, 296 plasma, 244 soliton, 88 state antibonding, 30, 379 bonding, 30, 379 densities, 22, 406 localised , 41, 50-56 substrates (preparation), 211 system degenerate, 103 host-guest, 362 non-degenerate, 105 thermoelectric power, 423 transition metal-insulator (Peierls), 24 metal-insulator (Anderson), 54 Mott-Hubbart, 44---46 permitted optical , 180 transition metals, 166, 171, 177 transport electron, 74, 99 polarons , 116 transport mechanisms, 99, 127, 133 traps, 144, 264, 463 distribution, 151 yield energetic, 274 external quantum , 221, 273 fluorescence quantum, 274 internal quantum, 221, 273 measure of, 226 photovoltaic, 333

497

Springer Series in

OPTICAL SCIENCES Volume 1 1 Solid-StateLaser Engineering ByW. Koechner, 5th revised and updated ed. 1999, 472 figs., 55 tabs., XII,746pages

Published titles sincevolume 80 85 Sensingwith Terahertz Radiation ByD.Mittleman (Ed.), 2003, 207figs., 14 tabs., XVI, 337 pages 86 Progress in Nano-Electro-Optics I Basics and Theory of Near-Field Optics ByM. Ohtsu (Ed.), 2003, us figs., XIV, 161 pages 87 Optical Imaging and Microscopy Techniques and Advanced Systems ByP. Torok, F.-J. Kao (Eds.), 2003, 260figs., XVII, 395 pages 88 Optical Interference Coatings ByN. Kaiser, H.K. Pulker (Eds.),2003, 203 figs., 50 tabs., XVI, 504pages 89 Progress in Nano-Electro-Optics II Novel Devices and AtomManipulation ByM. Ohtsu (Ed.), 2003, U5 figs., XIII,188 pages 90/1 Raman Amplifiersfor Telecommunications 1 Physical Principles ByM.N. Islam (Ed.), 2004, 488figs., XXVIII, 328 pages 90/2 Raman Amplifiersfor Telecommunications 2 Sub-Systems and Systems ByM.N. Islam (Ed.), 2004,278 figs., XXVIII, 420 pages 91 Optical Super Resolution ByZ. Zalevsky, D.Mendlovic, 2004,164 figs., XVIII, 232 pages 92 UV-VisibleReflection SpectroscopyofLiquids ByJ.A. Raty, K.-E. Peiponen,T.Asakura,2004,131 figs., XII,219 pages 93 Fundamentals of Semiconductor Lasers ByT.Numai,2004,166 figs., XII,264pages 94 Photonic Crystals Physics, Fabricationand Applications ByK. Inoue, K. Ohtaka (Eds.),2004, 209figs., XV, 320pages 95 Ultrafast Optics IV SelectedContributions to the 4th International Conference on UltrafastOptics, Vienna,Austria ByF. Krausz, G. Korn, P. Corkum, LA. Walmsley (Eds.),2004,281 figs., XIV, 506pages 96 Progress in Nano-ElectroOptics III Industrial Applications and Dynamicsof the Nano-Optical System ByM. Ohtsu (Bd.), 2004,186 figs., 8 tabs., XIV, 224 pages 97 Microoptics From Technology to Applications ByJ. Jahns, K.-H. Brenner,2004,303 figs., XI,335 pages 98 X-Ray Optics High-Energy-Resolution Applications ByY. Shvyd'ko, 2004,181 figs., XIV, 404 pages 99 Mono-Cycle Photonics and OpticalScanning TunnelingMicroscopy Routeto FemtosecondAngstromTechnology ByM. Yamashita, H. Shigekawa, R. Morita (Eds.)2005, 241 figs., XX, 393 pages 100 Quantum Interference and Coherence Theory and Experiments ByZ. Ficekand S. Swain, 2005, 178 figs., XV, 418 pages

SpringerSeries in

OPTICAL SCIENCES 101

Polarization Optics in Telecommunications By J. Damask, 2005, 110 figs., XVI, 528 pages

102

Lidar Range-Resolved OpticalRemote Sensingof the Atmosphere ByC. Weitkamp (Ed.), 161 figs., XX, 416 pages

103

OpticalFiber Fusion Splicing ByA.D. Yablon, 2005, 137 figs., XIII, 306 pages

104

Optoelectronicsof Molecules and Polymers ByA.Moliton, 2005, 229 figs., 592 pages

105

Solid-State Random Lasers ByM.Noginov, 2005, 131 figs., XII, 238 pages

106

CoherentSourcesofXUVRadiation SoftX-Ray Lasers and High-Order Harmonic Generation By P. Iaegle, 2005, 150 figs., approx. 264 pages

107

Optical Frequency-Modulated Continuous-Wave (FMCW) Interferometry ByJ. Zheng, 2005, 137 figs., XVIII, 254 pages

108

Laser Resonatorsand BeamPropagation Fundamentals, Advanced Concepts and Applications ByN. Hodgsonand H. Weber, 2005, 497 figs., approx. 790 pages

109

Progress in Nano-Electro Optics IV Characterization ofNano-Optical Materials and Optical Near-FieldInteractions ByM. Ohtsu (Ed.), 2005, 123 figs., XIV, 206 pages

110

Kramers-Kronig Relationsin OpticalMaterialsResearch ByV. Lucarini, J.J. Saarinen, K.-E. Peiponen, E.M. Vartiainen, 2005, 37 figs., X, 162 pages

111

SemiconductorLasers Stability, Instabilityand Chaos ByJ.Ohtsubo, 2005, 169 figs., XII, 438 pages

112

Photovoltaic Solar EnergyGeneration ByA. Goetzberger and v.u. Hoffmann, 2005, 139 figs., XII, 234 pages

113

PhotorefractiveMaterialsand Their Applications 1 BasicEffects ByP. Gunterand J.P. Huignard, 2005, 169 figs., approx.XII, 300 pages

114

PhotorefractiveMaterialsand Their Applications 2 Materials ByP. Gunter and J.P. Huignard, 2005, 100 figs., approx. XII,300 pages

115

PhotorefractiveMaterialsand Their Applications 3 Applications ByP. GUnter and J.P. Huignard, 2005, 100 figs., approx. XII,300 pages

116

Spatial Filtering Velocimetry Fundamentals and Applications By Y. Aizuand T. Asakura, 2006, 112 figs., approx.XII, 220 pages

Mid-infrared Semiconductor Optoelectronics (Springer Series in Optical Sciences)

Read more

Mid-infrared Semiconductor Optoelectronics (Springer Series in Optical Sciences)

Read more

Integrated Ring Resonators: The Compendium (Springer Series in Optical Sciences) (Springer Series in Optical Sciences)

Read more

Concentrator Photovoltaics (Springer Series in Optical Sciences)

Read more

Integrated Ring Resonators: The Compendium (Springer Series in Optical Sciences) (Springer Series in Optical Sciences)

Read more

Handbook of Nonlinear Optical Crystals (Springer Series in Optical Sciences)

Read more

Handbook of Nonlinear Optical Crystals (Springer Series in Optical Sciences)

Read more

Optical Interconnects : The Silicon Approach (Springer Series in Optical Sciences)

Read more

Optical Fiber Fusion Splicing (Springer Series in Optical Sciences)

Read more

Optical Fiber Fusion Splicing (Springer Series in Optical Sciences)

Read more

Optical Interconnects : The Silicon Approach (Springer Series in Optical Sciences)

Read more

Optical Interconnects : The Silicon Approach (Springer Series in Optical Sciences)

Read more

Fundamentals of Semiconductor Lasers (Springer Series in Optical Sciences)

Read more

Polarization Optics in Telecommunications (Springer Series in Optical Sciences)

Read more

Polarization Optics in Telecommunications (Springer Series in Optical Sciences)

Read more

Industrial Color Physics (Springer Series in Optical Sciences)

Read more

Industrial Color Physics (Springer Series in Optical Sciences)

Read more

Surface Plasmon Nanophotonics (Springer Series in Optical Sciences)

Read more

Engineering Optics, Third Edition (Springer Series in Optical Sciences 35)

Read more

Industrial Color Physics (Springer Series in Optical Sciences)

Read more

Geometrical Charged-Particle Optics (Springer Series in Optical Sciences)

Read more

Single-Photon Imaging (Springer Series in Optical Sciences 160)

Read more

Confocal Raman Microscopy (Springer Series in Optical Sciences)

Read more

Strong Field Laser Physics (Springer Series in Optical Sciences)

Read more

Optical Polarization of Molecules

Read more

Strong Field Laser Physics (Springer Series in Optical Sciences)

Read more

Single-Photon Imaging (Springer Series in Optical Sciences 160)

Read more

Engineering Optics, Third Edition (Springer Series in Optical Sciences 35)

Read more

Solid-State Laser Engineering (Springer Series in Optical Sciences)

Read more

Engineering Optics, Third Edition (Springer Series in Optical Sciences 35)

Read more

Recommend Documents

Mid-infrared Semiconductor Optoelectronics (Springer Series in Optical Sciences)

Springer Series in OPTICAL SCIENCES founded by H.K.V. Lotsch Editor-in-Chief: W.T. Rhodes, Atlanta Editorial Board: T....

Mid-infrared Semiconductor Optoelectronics (Springer Series in Optical Sciences)

Springer Series in OPTICAL SCIENCES founded by H.K.V. Lotsch Editor-in-Chief: W.T. Rhodes, Atlanta Editorial Board: T....

Integrated Ring Resonators: The Compendium (Springer Series in Optical Sciences) (Springer Series in Optical Sciences)

Springer Series in optical sciences founded by H.K.V. Lotsch Editor-in-Chief: W. T. Rhodes, Atlanta Editorial Board: A....

Concentrator Photovoltaics (Springer Series in Optical Sciences)

Springer Series in optical sciences founded by H.K.V. Lotsch Editor-in-Chief: W. T. Rhodes, Atlanta Editorial Board: A....

Integrated Ring Resonators: The Compendium (Springer Series in Optical Sciences) (Springer Series in Optical Sciences)

Springer Series in optical sciences founded by H.K.V. Lotsch Editor-in-Chief: W. T. Rhodes, Atlanta Editorial Board: A....

Handbook of Nonlinear Optical Crystals (Springer Series in Optical Sciences)

OPTICAL SCIENCES V. G. Dmitriev G.G. Gurzadyan D.N.Nikogosyan db k Han 00 of Nonlinear Optical Crystals Third Revis...

Handbook of Nonlinear Optical Crystals (Springer Series in Optical Sciences)

OPTICAL SCIENCES V. G. Dmitriev G.G. Gurzadyan D.N.Nikogosyan db k Han 00 of Nonlinear Optical Crystals Third Revis...

Optical Interconnects : The Silicon Approach (Springer Series in Optical Sciences)

Optical Fiber Fusion Splicing (Springer Series in Optical Sciences)

Springer Series in optical sciences founded by H.K.V. Lotsch Editor-in-Chief: W. T. Rhodes, Atlanta Editorial Board: T...

Optical Fiber Fusion Splicing (Springer Series in Optical Sciences)

Springer Series in optical sciences founded by H.K.V. Lotsch Editor-in-Chief: W. T. Rhodes, Atlanta Editorial Board: T...